Evolution algebras

Evolution Algeb ras Presented b y: Y olanda Cab rera Casado A dvisors: Mercedes Siles Molina, M.Victo ria V elasco Collado Universidad de M´ alaga 16 Decemb er 2017 Evolution Algeb ras Outline 1 Intro duction 2 Basic facts ab ou t evolution algeb ras Evolution algeb ras Pro duct and Change of basis Subalgeb ras and ideals Non-degenerate evolution algeb r a s The graph asso c ia t ed to an evolution algeb ra 3 Decomp osition of an evolution algeb ra Ideals generated b y one element Simple evolution algeb ras Reducible evolution algeb r a s The optimal direct-sum decomp osition of an evolution algeb ra 4 Classification t w o-dimensional evolution algeb ras 5 Classification of three-dimensional evolution algeb ras 6 F urther w o rk Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Outline 1 Intro duction 2 Basic facts ab out evolution algeb ras Evolution algeb ras Pro duct and Change of basis Subalgeb ras and ideals Non-degenerate evolution algeb r a s The graph asso c ia t ed to an evolution algeb ra 3 Decomp osition of an evolution algeb ra Ideals generated b y one element Simple evolution algeb ras Reducible evolution algeb r a s The optimal direct-sum decomp osition of an evolution algeb ra 4 Classification t w o-dimensional evolution algeb ras 5 Classification of three-dimensional evolution algeb ras 6 F urther w o rk Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Outline 1 Intro duction 2 Basic facts ab out evolution algeb ras Evolution algeb ras Pro duct and Change of basis Subalgeb ras and ideals Non-degenerate evolution algeb r a s The graph asso c ia t ed to an evolution algeb ra 3 Decomp osition of an evolution algeb ra Ideals generated b y one element Simple evolution algeb ras Reducible evolution algeb r a s The optimal direct-sum decomp osition of an evolution algeb ra 4 Classification t w o-dimensional evolution algeb ras 5 Classification of three-dimensional evolution algeb ras 6 F urther w o rk Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Outline 1 Intro duction 2 Basic facts ab out evolution algeb ras Evolution algeb ras Pro duct and Change of basis Subalgeb ras and ideals Non-degenerate evolution algeb r a s The graph asso c ia t ed to an evolution algeb ra 3 Decomp osition of an evolution algeb ra Ideals generated b y one element Simple evolution algeb ras Reducible evolution algeb r a s The optimal direct-sum decomp osition of an evolution algeb ra 4 Classification t w o-dimensional evolution algeb ras 5 Classification of three-dimensional evolution algeb ras 6 F urther w o rk Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Outline 1 Intro duction 2 Basic facts ab out evolution algeb ras Evolution algeb ras Pro duct and Change of basis Subalgeb ras and ideals Non-degenerate evolution algeb r a s The graph asso c ia t ed to an evolution algeb ra 3 Decomp osition of an evolution algeb ra Ideals generated b y one element Simple evolution algeb ras Reducible evolution algeb r a s The optimal direct-sum decomp osition of an evolution algeb ra 4 Classification t w o-dimensional evolution algeb ras 5 Classification of three-dimensional evolution algeb ras 6 F urther w o rk Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Outline 1 Intro duction 2 Basic facts ab out evolution algeb ras Evolution algeb ras Pro duct and Change of basis Subalgeb ras and ideals Non-degenerate evolution algeb ras The graph asso c ia t ed to an evolution algeb ra 3 Decomp osition of an evolution algeb ra Ideals generated b y one element Simple evolution algeb ras Reducible evolution algeb r a s The optimal direct-sum decomp osition of an evolution algeb ra 4 Classification t w o-dimensional evolution algeb ras 5 Classification of three-dimensional evolution algeb ras 6 F urther w o rk Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Outline 1 Intro duction 2 Basic facts ab out evolution algeb ras Evolution algeb ras Pro duct and Change of basis Subalgeb ras and ideals Non-degenerate evolution algeb ras The graph asso ciated to an evolution algeb ra 3 Decomp osition of an evolution algeb ra Ideals generated b y one element Simple evolution algeb ras Reducible evolution algeb r a s The optimal direct-sum decomp osition of an evolution algeb ra 4 Classification t w o-dimensional evolution algeb ras 5 Classification of three-dimensional evolution algeb ras 6 F urther w o rk Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Outline 1 Intro duction 2 Basic facts ab out evolution algeb ras Evolution algeb ras Pro duct and Change of basis Subalgeb ras and ideals Non-degenerate evolution algeb ras The graph asso ciated to an evolution algeb ra 3 Decomp osition of an evolution algeb ra Ideals generated b y one element Simple evolution algeb ras Reducible evolution algeb r a s The optimal direct-sum decomp osition of an evolution algeb ra 4 Classification t w o-dimensional evolution algeb ras 5 Classification of three-dimensional evolution algeb ras 6 F urther w o rk Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Outline 1 Intro duction 2 Basic facts ab out evolution algeb ras Evolution algeb ras Pro duct and Change of basis Subalgeb ras and ideals Non-degenerate evolution algeb ras The graph asso ciated to an evolution algeb ra 3 Decomp osition of an evolution algeb ra Ideals generated b y one element Simple evolution algeb ras Reducible evolution algeb ras The optimal direct-sum decomp osition of an evolution algeb ra 4 Classification t w o-dimensional evolution algeb ras 5 Classification of three-dimensional evolution algeb ras 6 F urther w o rk Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Intro duction Outline 1 Intro duction 2 Basic facts ab out evolution algeb ras Evolution algeb ras Pro duct and Change of basis Subalgeb ras and ideals Non-degenerate evolution algeb ras The graph asso ciated to an evolution algeb ra 3 Decomp osition of an evolution algeb ra Ideals generated b y one element Simple evolution algeb ras Reducible evolution algeb ras The optimal direct-sum decomp osition of an evolution algeb ra 4 Classification t w o-dimensional evolution algeb ras 5 Classification of three-dimensional evolution algeb ras 6 F urther w o rk Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 HIST OR Y OF AD V AN CES Me ndel p ublish es h is wo rk describing the b asi c laws o f inh eritanc e . 1865 Ser eb rowsky gave an algeb raic interpretatio n o f sign “x”. 1934 1939 Etherington p rov id ed a p rec ise math ematical f o rmulatio n o f M en del’s laws in ter ms o f nonasso ciativ e algeb ras. G livenkov introd uced the so - cal led M en delian algeb ras. 1936 G rego r Men d el, “ th e fath er o f geneti cs ”, begins d eta iled exp eri me nts breed ing p ea p lants . 1856 Evolution Algeb ras Intro duction Gene: Molecula r unit of heredita ry info rmation. Chromosomes: Long strands of DNA fo rmed b y o rdered se- quences of genes. In the p ro cess of rep ro duction, the attri- butes of the offs p ring a re inherited from alleles containe d in the chromosomes of the pa rents. Allele: Distinct fo r m s of genes to an attribute. F o r example, the gene fo r ey e colo r has three alleles: b ro wn, green and blue. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Intro duction Gene: Molecula r unit of heredita ry info rmation. Chromosomes: Long strands of DNA fo rmed b y o rdered se- quences of genes. In the p ro cess of rep ro duction, the attri- butes of the offs p ring a re inherited from alleles containe d in the chromosomes of the pa rents. Allele: Distinct fo r m s of genes to an attribute. F o r example, the gene fo r ey e colo r has three alleles: b ro wn, green and blue. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Intro duction Gene: Molecula r unit of heredita ry info rmation. Chromosomes: Long strands of DNA fo rmed b y o rdered se- quences of genes. In the p ro cess of rep ro duction, the attri- butes of the offsp ring a re inherited from alleles contained in the chromosomes of the pa rents. Allele: Distinct fo r m s of genes to an attribute. F o r example, the gene fo r ey e colo r has three alleles: b ro wn, green and blue. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Intro duction Gene: Molecula r unit of heredita ry info rmation. Chromosomes: Long strands of DNA fo rmed b y o rdered se- quences of genes. In the p ro cess of rep ro duction, the attri- butes of the offsp ring a re inherited from alleles contained in the chromosomes of the pa rents. Allele: Distinct fo rms of genes to an attribute. F o r example, the gene fo r ey e colo r has three alleles: b ro wn, green and blue. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Intro duction Diploid o rganisms ca rry a double set of chromo- somes (one of each pa rent). Otherwise it is called haploid. They rep ro duce b y means of sex cells (gametes) , each of them ca rrying a s in gle set of chromosomes. The fusion of t w o gametes of opp osite s e x gives rise to a zygote , which contains a double set of chromosomes. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Intro duction Diploid o rganisms ca rry a double set of chromo- somes (one of each pa rent). Otherwise it is called haploid. They rep ro duce b y means of sex cells (gametes) , each of them ca rrying a s in gle set of chromosomes. The fusion of t w o gametes of opp osite s e x gives rise to a zygote , which contains a double set of chromosomes. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Intro duction Diploid o rganisms ca rry a double set of chromo- somes (one of each pa rent). Otherwise it is called haploid. They rep ro duce b y means of sex cells (gametes) , each of them ca rrying a single set of chromosomes. The fusion of t w o gametes of opp osite s e x gives rise to a zygote , which contains a double set of chromosomes. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Intro duction Gametic Algeb ra The gametic algeb ra fo r simple Mendelian inheritance with t w o alleles { B , b } B b B B 1 2 (B+b) b 1 2 (B+b) b Consider the set of gametes B = { a 1 ,. .., a n } as abstract elements. Define th e n dimensional algeb ra over R with basis B and multipli- cation a i a j = n P k =1 γ ijk a k such that n P k =1 γ ijk = 1. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Intro duction Gametic Algeb ra The gametic algeb ra fo r simple Mendelian inheritance with t w o alleles { B , b } B b B B 1 2 (B+b) b 1 2 (B+b) b Consider the set of gametes B = { a 1 ,..., a n } as abstract elements. Define the n dimensional algeb ra over R with basis B and multipli- cation a i a j = n P k =1 γ ijk a k such that n P k =1 γ ijk = 1. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Intro duction P a rticula r case: evolution algeb ra In the asexual inheritance, • a i a j do es not mak e sense biologically ( a i a j = 0) i 6 = j . • a i a i = a 2 i = n P k =1 γ ki a k . Interp reted as self-replication. It is called evolution algeb ra. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Intro duction P a rticula r case: evolution algeb ra In the asexual inheritance, • a i a j do es not mak e sense biologically ( a i a j = 0) i 6 = j . • a i a i = a 2 i = n P k =1 γ ki a k . Interp reted as self-replication. It is called evolution algeb ra. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Intro duction P a rticula r case: evolution algeb ra In the asexual inheritance, • a i a j do es not mak e sense biologically ( a i a j = 0) i 6 = j . • a i a i = a 2 i = n P k =1 γ ki a k . Interp reted as self-replication. It is called evolution algeb ra. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Intro duction P a rticula r case: evolution algeb ra In the asexual inheritance, • a i a j do es not mak e sense biologically ( a i a j = 0) i 6 = j . • a i a i = a 2 i = n P k =1 γ ki a k . Interp reted as self-replication. It is called evolution algeb ra. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Basic facts ab out evolution algeb ras Outline 1 Intro duction 2 Basic facts ab out evolution algeb ras Evolution algeb ras Pro duct and Change of basis Subalgeb ras and ideals Non-degenerate evolution algeb ras The graph asso ciated to an evolution algeb ra 3 Decomp osition of an evolution algeb ra Ideals generated b y one element Simple evolution algeb ras Reducible evolution algeb ras The optimal direct-sum decomp osition of an evolution algeb ra 4 Classification t w o-dimensional evolution algeb ras 5 Classification of three-dimensional evolution algeb ras 6 F urther w o rk Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Basic facts ab out evolution algeb ras What w e mean b y evolution algeb ra Definitions An evolution algeb ra over a field K is a K -algeb ra A p rovided with a basis B = { e i | i ∈ Λ } such that e i e j = 0 whenever i 6 = j . • B is called a natural basis . • The scala rs ω ki ∈ K such that e 2 i := e i e i = P k ∈ Λ ω ki e k will b e called the structure constant s of A relative to B . • The matrix M B := ( ω ki ) is said to b e the structure mat r ix of A relative to B . Rema rk • |{ k ∈ Λ | ω ki 6 = 0 }| < ∞ fo r ever y i ∈ Λ ⇒ M B ∈ CFM Λ ( K ). • CFM Λ ( K )=( M Λ ( K ) , + , · ) such that fo r wh ich every column has at most a fin ite numb er of non-zero entries. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Basic facts ab out evolution algeb ras What w e mean b y evolution algeb ra Definitions An evolution algeb ra over a field K is a K -algeb ra A p rovided with a basis B = { e i | i ∈ Λ } such that e i e j = 0 whenever i 6 = j . • B is called a natural basis . • The scala rs ω ki ∈ K such that e 2 i := e i e i = P k ∈ Λ ω ki e k will b e called the structure constant s of A relative to B . • The matrix M B := ( ω ki ) is said to b e the structure mat r ix of A relative to B . Rema rk • |{ k ∈ Λ | ω ki 6 = 0 }| < ∞ fo r ever y i ∈ Λ ⇒ M B ∈ CFM Λ ( K ). • CFM Λ ( K )=( M Λ ( K ) , + , · ) such that fo r wh ich every column has at most a fin ite numb er of non-zero entries. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Basic facts ab out evolution algeb ras What w e mean b y evolution algeb ra Definitions An evolution algeb ra over a field K is a K -algeb ra A p rovided with a basis B = { e i | i ∈ Λ } such that e i e j = 0 whenever i 6 = j . • B is called a natural basis . • The scala rs ω ki ∈ K such that e 2 i := e i e i = P k ∈ Λ ω ki e k will b e called the structure constant s of A relative to B . • The matrix M B := ( ω ki ) is said to b e the structure mat r ix of A relative to B . Rema rk • |{ k ∈ Λ | ω ki 6 = 0 }| < ∞ fo r ever y i ∈ Λ ⇒ M B ∈ CFM Λ ( K ). • CFM Λ ( K )=( M Λ ( K ) , + , · ) such that fo r wh ich every column has at most a fin ite numb er of non-zero entries. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Basic facts ab out evolution algeb ras What w e mean b y evolution algeb ra Definitions An evolution algeb ra over a field K is a K -algeb ra A p rovided with a basis B = { e i | i ∈ Λ } such that e i e j = 0 whenever i 6 = j . • B is called a natural basis . • The scala rs ω ki ∈ K such that e 2 i := e i e i = P k ∈ Λ ω ki e k will b e called the structure constants of A relative to B . • The matrix M B := ( ω ki ) is said to b e the structure matrix of A relative to B . Rema rk • |{ k ∈ Λ | ω ki 6 = 0 }| < ∞ fo r ever y i ∈ Λ ⇒ M B ∈ CFM Λ ( K ). • CFM Λ ( K )=( M Λ ( K ) , + , · ) such that fo r wh ich every column has at most a fin ite numb er of non-zero entries. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Basic facts ab out evolution algeb ras What w e mean b y evolution algeb ra Definitions An evolution algeb ra over a field K is a K -algeb ra A p rovided with a basis B = { e i | i ∈ Λ } such that e i e j = 0 whenever i 6 = j . • B is called a natural basis . • The scala rs ω ki ∈ K such that e 2 i := e i e i = P k ∈ Λ ω ki e k will b e called the structure constants of A relative to B . • The matrix M B := ( ω ki ) is said to b e the structure matrix of A relative to B . Rema rk • |{ k ∈ Λ | ω ki 6 = 0 }| < ∞ fo r ever y i ∈ Λ ⇒ M B ∈ CFM Λ ( K ). • CFM Λ ( K )=( M Λ ( K ) , + , · ) such that fo r wh ich every column has at most a fin ite numb er of non-zero entries. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Basic facts ab out evolution algeb ras What w e mean b y evolution algeb ra Definitions An evolution algeb ra over a field K is a K -algeb ra A p rovided with a basis B = { e i | i ∈ Λ } such that e i e j = 0 whenever i 6 = j . • B is called a natural basis . • The scala rs ω ki ∈ K such that e 2 i := e i e i = P k ∈ Λ ω ki e k will b e called the structure constants of A relative to B . • The matrix M B := ( ω ki ) is said to b e the structure matrix of A relative to B . Rema rk • |{ k ∈ Λ | ω ki 6 = 0 }| < ∞ fo r ever y i ∈ Λ ⇒ M B ∈ CFM Λ ( K ). • CFM Λ ( K )=( M Λ ( K ) , + , · ) such that fo r wh ich every column has at most a fin ite numb er of non-zero entries. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Basic facts ab out evolution algeb ras What w e mean b y evolution algeb ra Definitions An evolution algeb ra over a field K is a K -algeb ra A p rovided with a basis B = { e i | i ∈ Λ } such that e i e j = 0 whenever i 6 = j . • B is called a natural basis . • The scala rs ω ki ∈ K such that e 2 i := e i e i = P k ∈ Λ ω ki e k will b e called the structure constants of A relative to B . • The matrix M B := ( ω ki ) is said to b e the structure matrix of A relative to B . Rema rk • |{ k ∈ Λ | ω ki 6 = 0 }| < ∞ fo r every i ∈ Λ ⇒ M B ∈ CFM Λ ( K ). • CFM Λ ( K )=( M Λ ( K ) , + , · ) such that fo r wh ich every column has at most a fin ite numb er of non-zero entries. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Basic facts ab out evolution algeb ras What w e mean b y evolution algeb ra Definitions An evolution algeb ra over a field K is a K -algeb ra A p rovided with a basis B = { e i | i ∈ Λ } such that e i e j = 0 whenever i 6 = j . • B is called a natural basis . • The scala rs ω ki ∈ K such that e 2 i := e i e i = P k ∈ Λ ω ki e k will b e called the structure constants of A relative to B . • The matrix M B := ( ω ki ) is said to b e the structure matrix of A relative to B . Rema rk • |{ k ∈ Λ | ω ki 6 = 0 }| < ∞ fo r every i ∈ Λ ⇒ M B ∈ CFM Λ ( K ). • CFM Λ ( K )=( M Λ ( K ) , + , · ) such that fo r wh ich every column has at most a fin ite numb er of non-zero entries. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Basic facts ab out evolution algeb ras What w e mean b y evolution algeb ra Definitions An evolution algeb ra over a field K is a K -algeb ra A p rovided with a basis B = { e i | i ∈ Λ } such that e i e j = 0 whenever i 6 = j . • B is called a natural basis . • The scala rs ω ki ∈ K such that e 2 i := e i e i = P k ∈ Λ ω ki e k will b e called the structure constants of A relative to B . • The matrix M B := ( ω ki ) is said to b e the structure matrix of A relative to B . Rema rk • |{ k ∈ Λ | ω ki 6 = 0 }| < ∞ fo r every i ∈ Λ ⇒ M B ∈ CFM Λ ( K ). • CFM Λ ( K )=( M Λ ( K ) , + , · ) such that fo r which every column has at most a finite numb er of non-zero entries. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Basic facts ab out evolution algeb ras Mendelian genetics versus non-Mendelian genetics Rema rk Let the zygotic algeb ra b e fo r simple Mendelian inheritance fo r one gene with t w o alleles, F and f . The zygotes have three p ossible genot yp es: FF , Ff and ff . W e c onsider vecto r space generate b y the basis B = { FF , Ff , ff } and acco rding to Mendel la ws the multiplication table is as follo ws: FF Ff ff FF FF 1 2 ( FF + Ff ) Ff Ff 1 2 ( FF + Ff ) 1 4 ( FF + ff ) + 1 2 Ff 1 2 ( ff + Ff ) ff Ff 1 2 ( ff + Ff ) ff Then this algeb ra is not an evolution algeb ra. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Basic facts ab out evolution algeb ras Mendelian genetics versus non-Mendelian genetics Rema rk Let the zygotic algeb ra b e fo r simple Mendelian inheritance fo r one gene with t w o alleles, F and f . The zygotes have three p ossible genot yp es: FF , Ff and ff . W e c onsider vecto r space generate b y the basis B = { FF , Ff , ff } and acco rding to Mendel la ws the multiplication table is as follo ws: FF Ff ff FF FF 1 2 ( FF + Ff ) Ff Ff 1 2 ( FF + Ff ) 1 4 ( FF + ff ) + 1 2 Ff 1 2 ( ff + Ff ) ff Ff 1 2 ( ff + Ff ) ff Then this algeb ra is not an evolution algeb ra. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Basic facts ab out evolution algeb ras Mendelian genetics versus non-Mendelian genetics Rema rk Let the zygotic algeb ra b e fo r simple Mendelian inheritance fo r one gene with t w o alleles, F and f . The zygotes have three p ossible genot yp es: FF , Ff and ff . W e c onsider vecto r space generate b y the basis B = { FF , Ff , ff } and acco rding to Mendel la ws the multiplication table is as follo ws: FF Ff ff FF FF 1 2 ( FF + Ff ) Ff Ff 1 2 ( FF + Ff ) 1 4 ( FF + ff ) + 1 2 Ff 1 2 ( ff + Ff ) ff Ff 1 2 ( ff + Ff ) ff Then this algeb ra is not an evolution algeb ra. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Basic facts ab out evolution algeb ras Mendelian genetics versus non-Mendelian genetics Rema rk Let the zygotic algeb ra b e fo r simple Mendelian inheritance fo r one gene with t w o alleles, F and f . The zygotes have three p ossible genot yp es: FF , Ff and ff . W e c onsider vecto r space generate b y the basis B = { FF , Ff , ff } and acco rding to Mendel la ws the multiplication table is as follo ws: FF Ff ff FF FF 1 2 ( FF + Ff ) Ff Ff 1 2 ( FF + Ff ) 1 4 ( FF + ff ) + 1 2 Ff 1 2 ( ff + Ff ) ff Ff 1 2 ( ff + Ff ) ff Then this algeb ra is not an evolution algeb ra. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Basic facts ab out evolution algeb ras Mendelian genetics versus non-Mendelian genetics Rema rk Let the zygotic algeb ra b e fo r simple Mendelian inheritance fo r one gene with t w o alleles, F and f . The zygotes have three p ossible genot yp es: FF , Ff and ff . W e consider vecto r space generate b y the basis B = { FF , Ff , ff } and acco rding to Mendel la ws the multiplication table is as follo ws: FF Ff ff FF FF 1 2 ( FF + Ff ) Ff Ff 1 2 ( FF + Ff ) 1 4 ( FF + ff ) + 1 2 Ff 1 2 ( ff + Ff ) ff Ff 1 2 ( ff + Ff ) ff Then this algeb ra is not an evolution algeb ra. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Basic facts ab out evolution algeb ras Mendelian genetics versus non-Mendelian genetics Rema rk Let the zygotic algeb ra b e fo r simple Mendelian inheritance fo r one gene with t w o alleles, F and f . The zygotes have three p ossible genot yp es: FF , Ff and ff . W e consider vecto r space generate b y the basis B = { FF , Ff , ff } and acco rding to Mendel la ws the multiplication table is as follo ws: FF Ff ff FF FF 1 2 ( FF + Ff ) Ff Ff 1 2 ( FF + Ff ) 1 4 ( FF + ff ) + 1 2 Ff 1 2 ( ff + Ff ) ff Ff 1 2 ( ff + Ff ) ff Then this algeb ra is not an evolution algebra. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Basic facts ab out evolution algeb ras Mendelian genetics versus non-Mendelian genetics Rema rk Let the zygotic algeb ra b e fo r simple Mendelian inheritance fo r one gene with t w o alleles, F and f . The zygotes have three p ossible genot yp es: FF , Ff and ff . W e consider vecto r space generate b y the basis B = { FF , Ff , ff } and acco rding to Mendel la ws the multiplication table is as follo ws: FF Ff ff FF FF 1 2 ( FF + Ff ) Ff Ff 1 2 ( FF + Ff ) 1 4 ( FF + ff ) + 1 2 Ff 1 2 ( ff + Ff ) ff Ff 1 2 ( ff + Ff ) ff Then this algeb ra is not an evolution algebra. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Basic facts ab out evolution algeb ras Prop erties Rema rk • Evolution algeb ras a re commutative and hence flexible. • The direct sum of evolution algeb ras is an evolution algeb ra. • The quotient algeb ra A / I with I ideal of A is an evolution algeb ra. • Evolution algeb ras a re not p o w er asso ciative in ge neral and therefo re it a re not, in general, Jo rdan, alternative o r asso ciative algeb ras. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Basic facts ab out evolution algeb ras Prop erties Rema rk • Evolution algeb ras a re commutative and hence flexible. • The direct sum of evolution algeb ras is an evolution algeb ra. • The quotient algeb ra A / I with I ideal of A is an evolution algeb ra. • Evolution algeb ras a re not p o w er asso ciative in ge neral and therefo re it a re not, in general, Jo rdan, alternative o r asso ciative algeb ras. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Basic facts ab out evolution algeb ras Prop erties Rema rk • Evolution algeb ras a re commutative and hence flexible. • The direct sum of evolution algeb ras is an evolution algeb ra. • The quotient algeb ra A / I with I ideal of A is an evolution algeb ra. • Evolution algeb ras a re not p o w er asso ciative in ge neral and therefo re it a re not, in general, Jo rdan, alternative o r asso ciative algeb ras. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Basic facts ab out evolution algeb ras Prop erties Rema rk • Evolution algeb ras a re commutative and hence flexible. • The direct sum of evolution algeb ras is an evolution algeb ra. • The quotient algeb ra A / I with I ideal of A is an evolution algeb ra. • Evolution algeb ras a re not p o w er asso ciative in ge neral and therefo re it a re not, in general, Jo rdan, alternative o r asso ciative algeb ras. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Basic facts ab out evolution algeb ras Prop erties Rema rk • Evolution algeb ras a re commutative and hence flexible. • The direct sum of evolution algeb ras is an evolution algeb ra. • The quotient algeb ra A / I with I ideal of A is an evolution algeb ra. • Evolution algeb ras a re not p o w er asso ciative in ge neral and therefo re it a re not, in general, Jo rdan, alternative o r asso ciative algeb ras. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Basic facts ab out evolution algeb ras Prop erties Rema rk • Evolution algeb ras a re commutative and hence flexible. • The direct sum of evolution algeb ras is an evolution algeb ra. • The quotient algeb ra A / I with I ideal of A is an evolution algeb ra. • Evolution algeb ras a re not p o w er asso ciative in ge neral and therefo re it a re not, in general, Jo rdan, alternative o r asso ciative algeb ras. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Basic facts ab out evolution algeb ras Prop erties Rema rk • Evolution algeb ras a re commutative and hence flexible. • The direct sum of evolution algeb ras is an evolution algeb ra. • The quotient algeb ra A / I with I ideal of A is an evolution algeb ra. • Evolution algeb ras a re not p o w er asso ciative in general and therefo re it a re not, in general, Jo rdan, alternative o r asso ciative algeb ras. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Basic facts ab out evolution algeb ras A new p ro duct Rema rk Let A b e any finite dimensional evolution algeb ra with a natural basis B . Supp ose x = P i ∈ Λ α i e i and y = P i ∈ Λ β i e i a rbitrary elements of A . Then, w e have ξ B ( xy ) = M B    α 1 β 1 . . . α n β n    . Definition Let A an evolution algeb ra and B = { e i | i ∈ Λ } a natural basis of A . F o r a rbitra ry ele m e nts x = P i ∈ Λ x α i e i and y = P i ∈ Λ y β i e i in A fo r certain Λ x , Λ y ⊆ Λ, w e define x • B y := X i ∈ Λ x ∩ Λ y α i β i e i . Rema rk ξ B ( xy ) = M B ( ξ B ( x ) • B ξ B ( y )) , Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Basic facts ab out evolution algeb ras A new p ro duct Rema rk Let A b e any finite dimensional evolution algeb ra with a natural basis B . Supp ose x = P i ∈ Λ α i e i and y = P i ∈ Λ β i e i a rbitrary elements of A . Then, w e have ξ B ( xy ) = M B    α 1 β 1 . . . α n β n    . Definition Let A an evolution algeb ra and B = { e i | i ∈ Λ } a natural basis of A . F o r a rbitra ry ele m e nts x = P i ∈ Λ x α i e i and y = P i ∈ Λ y β i e i in A fo r certain Λ x , Λ y ⊆ Λ, w e define x • B y := X i ∈ Λ x ∩ Λ y α i β i e i . Rema rk ξ B ( xy ) = M B ( ξ B ( x ) • B ξ B ( y )) , Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Basic facts ab out evolution algeb ras A new p ro duct Rema rk Let A b e any finite dimensional evolution algeb ra with a natural basis B . Supp ose x = P i ∈ Λ α i e i and y = P i ∈ Λ β i e i a rbitrary elements of A . Then, w e have ξ B ( xy ) = M B    α 1 β 1 . . . α n β n    . Definition Let A an evolution algeb ra and B = { e i | i ∈ Λ } a natural basis of A . F o r a rbitrary elements x = P i ∈ Λ x α i e i and y = P i ∈ Λ y β i e i in A fo r certain Λ x , Λ y ⊆ Λ, w e define x • B y := X i ∈ Λ x ∩ Λ y α i β i e i . Rema rk ξ B ( xy ) = M B ( ξ B ( x ) • B ξ B ( y )) , Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Basic facts ab out evolution algeb ras A new p ro duct Rema rk Let A b e any finite dimensional evolution algeb ra with a natural basis B . Supp ose x = P i ∈ Λ α i e i and y = P i ∈ Λ β i e i a rbitrary elements of A . Then, w e have ξ B ( xy ) = M B    α 1 β 1 . . . α n β n    . Definition Let A an evolution algeb ra and B = { e i | i ∈ Λ } a natural basis of A . F o r a rbitrary elements x = P i ∈ Λ x α i e i and y = P i ∈ Λ y β i e i in A fo r certain Λ x , Λ y ⊆ Λ, w e define x • B y := X i ∈ Λ x ∩ Λ y α i β i e i . Rema rk ξ B ( xy ) = M B ( ξ B ( x ) • B ξ B ( y )) , Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Basic facts ab out evolution algeb ras A new p ro duct Rema rk Let A b e any finite dimensional evolution algeb ra with a natural basis B . Supp ose x = P i ∈ Λ α i e i and y = P i ∈ Λ β i e i a rbitrary elements of A . Then, w e have ξ B ( xy ) = M B    α 1 β 1 . . . α n β n    . Definition Let A an evolution algeb ra and B = { e i | i ∈ Λ } a natural basis of A . F o r a rbitrary elements x = P i ∈ Λ x α i e i and y = P i ∈ Λ y β i e i in A fo r certain Λ x , Λ y ⊆ Λ, w e define x • B y := X i ∈ Λ x ∩ Λ y α i β i e i . Rema rk ξ B ( xy ) = M B ( ξ B ( x ) • B ξ B ( y )) , Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Basic facts ab out evolution algeb ras A new p ro duct Rema rk Let A b e any finite dimensional evolution algeb ra with a natural basis B . Supp ose x = P i ∈ Λ α i e i and y = P i ∈ Λ β i e i a rbitrary elements of A . Then, w e have ξ B ( xy ) = M B    α 1 β 1 . . . α n β n    . Definition Let A an evolution algeb ra and B = { e i | i ∈ Λ } a natural basis of A . F o r a rbitrary elements x = P i ∈ Λ x α i e i and y = P i ∈ Λ y β i e i in A fo r certain Λ x , Λ y ⊆ Λ, w e define x • B y := X i ∈ Λ x ∩ Λ y α i β i e i . Rema rk ξ B ( xy ) = M B ( ξ B ( x ) • B ξ B ( y )) , Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Basic facts ab out evolution algeb ras Change of basis Theo rem Let A b e an evolution a lge b ra and let B = { e i | i ∈ Λ } b e a natural basis of A with structure ma tr ix M B = ( ω ij ). 1 If B 0 = { f i | i ∈ Λ } is a natural basis of A with P B 0 B = ( p ij ) the change of basis matrices, the n M B ( ξ B ( f i ) • B ξ B ( f j )) = 0 fo r e very i 6 = j with i , j ∈ Λ. Mo reover M B 0 = P − 1 B 0 B M B P (2) B 0 B , where P (2) B 0 B = ( p 2 ij ) . 2 Assume that P = ( p ij ) ∈ CFM Λ ( K ) is invertible and satisfies the first ab ove relation. Define B 0 = { f i | i ∈ Λ } , where f i = P j ∈ Λ p ji e j fo r e very i ∈ Λ. Then B 0 is a natural basis and the second ab ove relation is satisfied. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Basic facts ab out evolution algeb ras Change of basis Theo rem Let A b e an evolution a lge b ra and let B = { e i | i ∈ Λ } b e a natural basis of A with structure ma tr ix M B = ( ω ij ). 1 If B 0 = { f i | i ∈ Λ } is a natural basis of A with P B 0 B = ( p ij ) the change of basis matrices, the n M B ( ξ B ( f i ) • B ξ B ( f j )) = 0 fo r e very i 6 = j with i , j ∈ Λ. Mo reover M B 0 = P − 1 B 0 B M B P (2) B 0 B , where P (2) B 0 B = ( p 2 ij ) . 2 Assume that P = ( p ij ) ∈ CFM Λ ( K ) is invertible and satisfies the first ab ove relation. Define B 0 = { f i | i ∈ Λ } , where f i = P j ∈ Λ p ji e j fo r e very i ∈ Λ. Then B 0 is a natural basis and the second ab ove relation is satisfied. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Basic facts ab out evolution algeb ras Change of basis Theo rem Let A b e an evolution algeb ra and let B = { e i | i ∈ Λ } b e a natural basis of A with structure matrix M B = ( ω ij ). 1 If B 0 = { f i | i ∈ Λ } is a natural basis of A with P B 0 B = ( p ij ) the change of basis matrices, the n M B ( ξ B ( f i ) • B ξ B ( f j )) = 0 fo r e very i 6 = j with i , j ∈ Λ. Mo reover M B 0 = P − 1 B 0 B M B P (2) B 0 B , where P (2) B 0 B = ( p 2 ij ) . 2 Assume that P = ( p ij ) ∈ CFM Λ ( K ) is invertible and satisfies the first ab ove relation. Define B 0 = { f i | i ∈ Λ } , where f i = P j ∈ Λ p ji e j fo r e very i ∈ Λ. Then B 0 is a natural basis and the second ab ove relation is satisfied. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Basic facts ab out evolution algeb ras Change of basis Theo rem Let A b e an evolution algeb ra and let B = { e i | i ∈ Λ } b e a natural basis of A with structure matrix M B = ( ω ij ). 1 If B 0 = { f i | i ∈ Λ } is a natural basis of A with P B 0 B = ( p ij ) the change of basis matrices, then M B ( ξ B ( f i ) • B ξ B ( f j )) = 0 fo r e very i 6 = j with i , j ∈ Λ. Mo reover M B 0 = P − 1 B 0 B M B P (2) B 0 B , where P (2) B 0 B = ( p 2 ij ) . 2 Assume that P = ( p ij ) ∈ CFM Λ ( K ) is invertible and satisfies the first ab ove relation. Define B 0 = { f i | i ∈ Λ } , where f i = P j ∈ Λ p ji e j fo r e very i ∈ Λ. Then B 0 is a natural basis and the second ab ove relation is satisfied. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Basic facts ab out evolution algeb ras Change of basis Theo rem Let A b e an evolution algeb ra and let B = { e i | i ∈ Λ } b e a natural basis of A with structure matrix M B = ( ω ij ). 1 If B 0 = { f i | i ∈ Λ } is a natural basis of A with P B 0 B = ( p ij ) the change of basis matrices, then M B ( ξ B ( f i ) • B ξ B ( f j )) = 0 fo r every i 6 = j with i , j ∈ Λ. Mo reover M B 0 = P − 1 B 0 B M B P (2) B 0 B , where P (2) B 0 B = ( p 2 ij ) . 2 Assume that P = ( p ij ) ∈ CFM Λ ( K ) is invertible and satisfies the first ab ove relation. Define B 0 = { f i | i ∈ Λ } , where f i = P j ∈ Λ p ji e j fo r every i ∈ Λ. Then B 0 is a natural basis and the second ab ove relation is satisfied. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Basic facts ab out evolution algeb ras Change of basis Theo rem Let A b e an evolution algeb ra and let B = { e i | i ∈ Λ } b e a natural basis of A with structure matrix M B = ( ω ij ). 1 If B 0 = { f i | i ∈ Λ } is a natural basis of A with P B 0 B = ( p ij ) the change of basis matrices, then M B ( ξ B ( f i ) • B ξ B ( f j )) = 0 fo r every i 6 = j with i , j ∈ Λ. Mo reover M B 0 = P − 1 B 0 B M B P (2) B 0 B , where P (2) B 0 B = ( p 2 ij ) . 2 Assume that P = ( p ij ) ∈ CFM Λ ( K ) is invertible and satisfies the first ab ove relation. Define B 0 = { f i | i ∈ Λ } , where f i = P j ∈ Λ p ji e j fo r every i ∈ Λ. Then B 0 is a natural basis and the second ab ove relation is satisfied. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Basic facts ab out evolution algeb ras Evolution subalgeb ras. Evolution ideals Definitions • An subalgeb ra A 0 of an evolution algeb ra A with natural basis B = { e i | i ∈ Λ } is an evolution subalgeb ra if there exits a natural basis. • An ideal I of A is called evolution ideal if it is an evolution subalgeb ra. Rema rk 1 A subalgeb ra of an e volution algeb ra do es not need to b e an evolution algeb ra. 2 An evolution s u balgeb ra do es not need to b e an ideal. 3 Not every ideal of an e volution algeb ra has a natural basis. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Basic facts ab out evolution algeb ras Evolution subalgeb ras. Evolution ideals Definitions • An subalgeb ra A 0 of an evolution algeb ra A with natural basis B = { e i | i ∈ Λ } is an evolution subalgeb ra if there exits a natural basis. • An ideal I of A is called evolution ideal if it is an evolution subalgeb ra. Rema rk 1 A subalgeb ra of an e volution algeb ra do es not need to b e an evolution algeb ra. 2 An evolution s u balgeb ra do es not need to b e an ideal. 3 Not every ideal of an e volution algeb ra has a natural basis. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Basic facts ab out evolution algeb ras Evolution subalgeb ras. Evolution ideals Definitions • An subalgeb ra A 0 of an evolution algeb ra A with natural basis B = { e i | i ∈ Λ } is an evolution subalgeb ra if there exits a natural basis. • An ideal I of A is called evolution ideal if it is an evolution subalgeb ra. Rema rk 1 A subalgeb ra of an e volution algeb ra do es not need to b e an evolution algeb ra. 2 An evolution s u balgeb ra do es not need to b e an ideal. 3 Not every ideal of an e volution algeb ra has a natural basis. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Basic facts ab out evolution algeb ras Evolution subalgeb ras. Evolution ideals Definitions • An subalgeb ra A 0 of an evolution algeb ra A with natural basis B = { e i | i ∈ Λ } is an evolution subalgeb ra if there exits a natural basis. • An ideal I of A is called evolution ideal if it is an evolution subalgeb ra. Rema rk 1 A subalgeb ra of an e volution algeb ra do es not need to b e an evolution algeb ra. 2 An evolution s u balgeb ra do es not need to b e an ideal. 3 Not every ideal of an e volution algeb ra has a natural basis. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Basic facts ab out evolution algeb ras Evolution subalgeb ras. Evolution ideals Definitions • An subalgeb ra A 0 of an evolution algeb ra A with natural basis B = { e i | i ∈ Λ } is an evolution subalgeb ra if there exits a natural basis. • An ideal I of A is called evolution ideal if it is an evolution subalgeb ra. Rema rk 1 A subalgeb ra of an e volution algeb ra do es not need to b e an evolution algeb ra. 2 An evolution s u balgeb ra do es not need to b e an ideal. 3 Not every ideal of an e volution algeb ra has a natural basis. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Basic facts ab out evolution algeb ras Evolution subalgeb ras. Evolution ideals Definitions • An subalgeb ra A 0 of an evolution algeb ra A with natural basis B = { e i | i ∈ Λ } is an evolution subalgeb ra if there exits a natural basis. • An ideal I of A is called evolution ideal if it is an evolution subalgeb ra . Rema rk 1 A subalgeb ra of an e volution algeb ra do es not need to b e an evolution algeb ra. 2 An evolution s u balgeb ra do es not need to b e an ideal. 3 Not every ideal of an e volution algeb ra has a natural basis. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Basic facts ab out evolution algeb ras Evolution subalgeb ras. Evolution ideals Definitions • An subalgeb ra A 0 of an evolution algeb ra A with natural basis B = { e i | i ∈ Λ } is an evolution subalgeb ra if there exits a natural basis. • An ideal I of A is called evolution ideal if it is an evolution subalgeb ra. Rema rk 1 A subalgeb ra of an e volution algeb ra do es not need to b e an evolution algeb ra. 2 An evolution s u balgeb ra do es not need to b e an ideal. 3 Not every ideal of an e volution algeb ra has a natural basis. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Basic facts ab out evolution algeb ras Evolution subalgeb ras. Evolution ideals Definitions • An subalgeb ra A 0 of an evolution algeb ra A with natural basis B = { e i | i ∈ Λ } is an evolution subalgeb ra if there exits a natural basis. • An ideal I of A is called evolution ideal if it is an evolution subalgeb ra. Rema rk 1 A subalgeb ra of an evolution algeb ra do es not need to b e an evolution algeb ra. 2 An evolution s u balgeb ra do es not need to b e an ideal. 3 Not every ideal of an e volution algeb ra has a natural basis. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Basic facts ab out evolution algeb ras Something less restrictive and mo re algeb raically natural Definition W e sa y that an subalgeb ra A 0 has the extension p rop ert y if there exists a natural basis B 0 of A 0 which can b e extended to a natural basis of A . Rema rk Not every evolu tion subalgeb ra has the extension p rop ert y . Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Basic facts ab out evolution algeb ras Something less restrictive and mo re algeb raically natural Definition W e sa y that an subalgeb ra A 0 has the extension p rop ert y if there exists a natural basis B 0 of A 0 which can b e extended to a natural basis of A . Rema rk Not every evolution subalgeb ra has the extension p rop ert y . Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Basic facts ab out evolution algeb ras Examples of evolution subalgeb ras. Homomo rphism Co rolla ry Let f : A → A 0 b e a homom o rphism b et w een the evolution algeb ras A and A 0 . Then Im ( f ) is an evolution subalgeb ra of A 0 . Rema rk In general, Ker ( f ) is not an evolution algeb ra. ⇒ [Evolution Algeb ras and their Applications, Theo r em 2, p.25] is not valid in general. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Basic facts ab out evolution algeb ras Examples of evolution subalgeb ras. Homomo rphism Co rolla ry Let f : A → A 0 b e a homom o rphism b et w een the evolution algeb ras A and A 0 . Then Im ( f ) is an evolution subalgeb ra of A 0 . Rema rk In general, Ker ( f ) is not an evolution algeb ra. ⇒ [Evolution Algeb ras and their Applications, Theo r em 2, p.25] is not valid in general. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Basic facts ab out evolution algeb ras Examples of evolution subalgeb ras. Homomo rphism Co rolla ry Let f : A → A 0 b e a homomo rphism b et w een the evolution algeb ras A and A 0 . Then Im ( f ) is an evolution subalgeb ra of A 0 . Rema rk In general, Ker ( f ) is not an evolution algeb ra. ⇒ [Evolution Algeb ras and their Applications, Theo r em 2, p.25] is not valid in general. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Basic facts ab out evolution algeb ras Examples of evolution subalgeb ras. Homomo rphism Co rolla ry Let f : A → A 0 b e a homomo rphism b et w een the evolution algeb ras A and A 0 . Then Im ( f ) is an evolution subalgeb ra of A 0 . Rema rk In general, Ker ( f ) is not an evolution algeb ra. ⇒ [Evolution Algeb ras and their Applications, Theo r em 2, p.25] is not valid in general. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Basic facts ab out evolution algeb ras Examples of evolution subalgeb ras. Homomo rphism Co rolla ry Let f : A → A 0 b e a homomo rphism b et w een the evolution algeb ras A and A 0 . Then Im ( f ) is an evolution subalgeb ra of A 0 . Rema rk In general, Ker ( f ) is not an evolution algeb ra. ⇒ [Evolution Algeb ras and their Applications, Theo r em 2, p.25] is not valid in general. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Basic facts ab out evolution algeb ras Examples of evolution subalgeb ras. Homomo rphism Co rolla ry Let f : A → A 0 b e a homomo rphism b et w een the evolution algeb ras A and A 0 . Then Im ( f ) is an evolution subalgeb ra of A 0 . Rema rk In general, Ker ( f ) is not an evolution algeb ra. ⇒ [Evolution Algeb ras and their Applications, Theo rem 2, p.25] is not valid in general. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Basic facts ab out evolution algeb ras Examples of evolution subalgeb ras. Homomo rphism Co rolla ry Let f : A → A 0 b e a homomo rphism b et w een the evolution algeb ras A and A 0 . Then Im ( f ) is an evolution subalgeb ra of A 0 . Rema rk In general, Ker ( f ) is not an evolution algeb ra . ⇒ [Evolution Algeb ras and their Applications, Theo rem 2, p.25] is not valid in general. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Basic facts ab out evolution algeb ras Non-degenerate evolution algeb ra Definition An evolution algeb ra A is non-degenerate if it has a n a tu ra l basis B = { e i | i ∈ Λ } such that e 2 i 6 = 0 fo r every i ∈ Λ. Rema rk Do es non-degeneracy dep end on the conside red natural basis? Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Basic facts ab out evolution algeb ras Annihilato r. Prop erties Definition Let A b e an commutative algeb ra, w e define its annihilato r as ann ( A ) := { x ∈ A | xA = 0 } . Prop osition Let A b e an evolu tion algeb ra and B = { e i | i ∈ Λ } a natural basis. Denote b y Λ 0 ( B ) := { i ∈ Λ | e 2 i = 0 } . Then 1 ann ( A ) = lin { e i ∈ B | i ∈ Λ 0 ( B ) } . 2 ann ( A ) = 0 if and on ly if Λ 0 = ∅ . 3 ann ( A ) is an evolution ideal of A . 4 | Λ 0 ( B ) | = | Λ 0 ( B 0 ) | fo r e very natural basis B 0 of A . Consequently , the definition of non-dege nerate evolution algeb ra do es not dep end on the considered natural basis. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Basic facts ab out evolution algeb ras Annihilato r. Prop erties Definition Let A b e an commutative algeb ra, w e define its annihilato r as ann ( A ) := { x ∈ A | xA = 0 } . Prop osition Let A b e an evolu tion algeb ra and B = { e i | i ∈ Λ } a natural basis. Denote b y Λ 0 ( B ) := { i ∈ Λ | e 2 i = 0 } . Then 1 ann ( A ) = lin { e i ∈ B | i ∈ Λ 0 ( B ) } . 2 ann ( A ) = 0 if and on ly if Λ 0 = ∅ . 3 ann ( A ) is an evolution ideal of A . 4 | Λ 0 ( B ) | = | Λ 0 ( B 0 ) | fo r e very natural basis B 0 of A . Consequently , the definition of non-dege nerate evolution algeb ra do es not dep end on the considered natural basis. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Basic facts ab out evolution algeb ras Annihilato r. Prop erties Definition Let A b e an commutative algeb ra, w e define its annihilato r as ann ( A ) := { x ∈ A | xA = 0 } . Prop osition Let A b e an evolu tion algeb ra and B = { e i | i ∈ Λ } a natural basis. Denote b y Λ 0 ( B ) := { i ∈ Λ | e 2 i = 0 } . Then 1 ann ( A ) = lin { e i ∈ B | i ∈ Λ 0 ( B ) } . 2 ann ( A ) = 0 if and on ly if Λ 0 = ∅ . 3 ann ( A ) is an evolution ideal of A . 4 | Λ 0 ( B ) | = | Λ 0 ( B 0 ) | fo r e very natural basis B 0 of A . Consequently , the definition of non-dege nerate evolution algeb ra do es not dep end on the considered natural basis. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Basic facts ab out evolution algeb ras Annihilato r. Prop erties Definition Let A b e an commutative algeb ra, w e define its annihilato r as ann ( A ) := { x ∈ A | xA = 0 } . Prop osition Let A b e an evolution algeb ra and B = { e i | i ∈ Λ } a natural basis. Denote b y Λ 0 ( B ) := { i ∈ Λ | e 2 i = 0 } . Then 1 ann ( A ) = lin { e i ∈ B | i ∈ Λ 0 ( B ) } . 2 ann ( A ) = 0 if and on ly if Λ 0 = ∅ . 3 ann ( A ) is an evolution ideal of A . 4 | Λ 0 ( B ) | = | Λ 0 ( B 0 ) | fo r e very natural basis B 0 of A . Consequently , the definition of non-dege nerate evolution algeb ra do es not dep end on the considered natural basis. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Basic facts ab out evolution algeb ras Annihilato r. Prop erties Definition Let A b e an commutative algeb ra, w e define its annihilato r as ann ( A ) := { x ∈ A | xA = 0 } . Prop osition Let A b e an evolution algeb ra and B = { e i | i ∈ Λ } a natural basis. Denote b y Λ 0 ( B ) := { i ∈ Λ | e 2 i = 0 } . Then 1 ann ( A ) = lin { e i ∈ B | i ∈ Λ 0 ( B ) } . 2 ann ( A ) = 0 if and on ly if Λ 0 = ∅ . 3 ann ( A ) is an evolution ideal of A . 4 | Λ 0 ( B ) | = | Λ 0 ( B 0 ) | fo r e very natural basis B 0 of A . Consequently , the definition of non-dege nerate evolution algeb ra do es not dep end on the considered natural basis. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Basic facts ab out evolution algeb ras Annihilato r. Prop erties Definition Let A b e an commutative algeb ra, w e define its annihilato r as ann ( A ) := { x ∈ A | xA = 0 } . Prop osition Let A b e an evolution algeb ra and B = { e i | i ∈ Λ } a natural basis. Denote b y Λ 0 ( B ) := { i ∈ Λ | e 2 i = 0 } . Then 1 ann ( A ) = lin { e i ∈ B | i ∈ Λ 0 ( B ) } . 2 ann ( A ) = 0 if and on ly if Λ 0 = ∅ . 3 ann ( A ) is an evolution ideal of A . 4 | Λ 0 ( B ) | = | Λ 0 ( B 0 ) | fo r e very natural basis B 0 of A . Consequently , the definition of non-dege nerate evolution algeb ra do es not dep end on the considered natural basis. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Basic facts ab out evolution algeb ras Annihilato r. Prop erties Definition Let A b e an commutative algeb ra, w e define its annihilato r as ann ( A ) := { x ∈ A | xA = 0 } . Prop osition Let A b e an evolution algeb ra and B = { e i | i ∈ Λ } a natural basis. Denote b y Λ 0 ( B ) := { i ∈ Λ | e 2 i = 0 } . Then 1 ann ( A ) = lin { e i ∈ B | i ∈ Λ 0 ( B ) } . 2 ann ( A ) = 0 if and on ly if Λ 0 = ∅ . 3 ann ( A ) is an evolution ideal of A . 4 | Λ 0 ( B ) | = | Λ 0 ( B 0 ) | fo r e very natural basis B 0 of A . Consequently , the definition of non-dege nerate evolution algeb ra do es not dep end on the considered natural basis. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Basic facts ab out evolution algeb ras Annihilato r. Prop erties Definition Let A b e an commutative algeb ra, w e define its annihilato r as ann ( A ) := { x ∈ A | xA = 0 } . Prop osition Let A b e an evolution algeb ra and B = { e i | i ∈ Λ } a natural basis. Denote b y Λ 0 ( B ) := { i ∈ Λ | e 2 i = 0 } . Then 1 ann ( A ) = lin { e i ∈ B | i ∈ Λ 0 ( B ) } . 2 ann ( A ) = 0 if and only if Λ 0 = ∅ . 3 ann ( A ) is an evolution ideal of A . 4 | Λ 0 ( B ) | = | Λ 0 ( B 0 ) | fo r e very natural basis B 0 of A . Consequently , the definition of non-dege nerate evolution algeb ra do es not dep end on the considered natural basis. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Basic facts ab out evolution algeb ras Annihilato r. Prop erties Definition Let A b e an commutative algeb ra, w e define its annihilato r as ann ( A ) := { x ∈ A | xA = 0 } . Prop osition Let A b e an evolution algeb ra and B = { e i | i ∈ Λ } a natural basis. Denote b y Λ 0 ( B ) := { i ∈ Λ | e 2 i = 0 } . Then 1 ann ( A ) = lin { e i ∈ B | i ∈ Λ 0 ( B ) } . 2 ann ( A ) = 0 if and only if Λ 0 = ∅ . 3 ann ( A ) is an evolution ideal of A . 4 | Λ 0 ( B ) | = | Λ 0 ( B 0 ) | fo r e very natural basis B 0 of A . Consequently , the definition of non-dege nerate evolution algeb ra do es not dep end on the considered natural basis. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Basic facts ab out evolution algeb ras Annihilato r. Prop erties Definition Let A b e an commutative algeb ra, w e define its annihilato r as ann ( A ) := { x ∈ A | xA = 0 } . Prop osition Let A b e an evolution algeb ra and B = { e i | i ∈ Λ } a natural basis. Denote b y Λ 0 ( B ) := { i ∈ Λ | e 2 i = 0 } . Then 1 ann ( A ) = lin { e i ∈ B | i ∈ Λ 0 ( B ) } . 2 ann ( A ) = 0 if and only if Λ 0 = ∅ . 3 ann ( A ) is an evolution ideal of A . 4 | Λ 0 ( B ) | = | Λ 0 ( B 0 ) | fo r every natural basis B 0 of A . Consequently , the definition of non-dege nerate evolution algeb ra do es not dep end on the considered natural basis. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Basic facts ab out evolution algeb ras Annihilato r. Prop erties Rema rk Let A b e an evolution algeb ra and B = { e i | i ∈ Λ } a natural basis. Denote b y Λ 1 := { i ∈ Λ | e 2 i 6 = 0 } . Then 1 A 1 := lin { e i ∈ B | i ∈ Λ 1 } is not necessa rily a subalgeb ra of A . 2 A / ann ( A ) is not necessa rily a non-degenerate evolution algeb ra. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Basic facts ab out evolution algeb ras Annihilato r. Prop erties Rema rk Let A b e an evolution algeb ra and B = { e i | i ∈ Λ } a natural basis. Denote b y Λ 1 := { i ∈ Λ | e 2 i 6 = 0 } . Then 1 A 1 := lin { e i ∈ B | i ∈ Λ 1 } is not necessa rily a subalgeb ra of A . 2 A / ann ( A ) is not necessa rily a non-degenerate evolution algeb ra . Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Basic facts ab out evolution algeb ras Abso rption p rop ert y . Prop erties Definition Let I b e an ideal of an evolution algeb ra A . I has the abso rption p r op ert y if xA ⊆ I implies x ∈ I . Lemma • An ideal I of an evolution algeb ra A has the abso rption p rop ert y if and only if ann ( A / I ) = 0. • If I is a non-zero ideal which it has the a b s o rption p rop ert y, then I is an evolution ideal and has the extension p rop ert y . Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Basic facts ab out evolution algeb ras Abso rption p rop ert y . Prop erties Definition Let I b e an ideal of an evolution algeb ra A . I has the abso rption p r op ert y if xA ⊆ I implies x ∈ I . Lemma • An ideal I of an evolution algeb ra A has the abso rption p rop ert y if and only if ann ( A / I ) = 0. • If I is a non-zero ideal which it has the a b s o rption p rop ert y, then I is an evolution ideal and has the extension p rop ert y . Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Basic facts ab out evolution algeb ras Abso rption p rop ert y . Prop erties Definition Let I b e an ideal of an evolution algeb ra A . I has the abso rption p r op ert y if xA ⊆ I implies x ∈ I . Lemma • An ideal I of an evolution algeb ra A has the abso rption p rop ert y if and only if ann ( A / I ) = 0. • If I is a non-zero ideal which it has the a b s o rption p rop ert y, then I is an evolution ideal and has the extension p rop ert y . Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Basic facts ab out evolution algeb ras Abso rption p rop ert y . Prop erties Definition Let I b e an ideal of an evolution algeb ra A . I has the abso rption p rop ert y if xA ⊆ I implies x ∈ I . Lemma • An ideal I of an evolution algeb ra A has the abso rption p rop ert y if and only if ann ( A / I ) = 0. • If I is a non-zero ideal which it has the a b s o rption p rop ert y, then I is an evolution ideal and has the extension p rop ert y . Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Basic facts ab out evolution algeb ras Abso rption p rop ert y . Prop erties Definition Let I b e an ideal of an evolution algeb ra A . I has the abso rption p rop ert y if xA ⊆ I implies x ∈ I . Lemma • An ideal I of an evolution algeb ra A has the abso rption p rop ert y if and only if ann ( A / I ) = 0. • If I is a non-zero ideal which it has the a b s o rption p rop ert y, then I is an evolution ideal and has the extension p rop ert y . Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Basic facts ab out evolution algeb ras Abso rption p rop ert y . Prop erties Definition Let I b e an ideal of an evolution algeb ra A . I has the abso rption p rop ert y if xA ⊆ I implies x ∈ I . Lemma • An ideal I of an evolution algeb ra A has the abso rption p rop ert y if and only if ann ( A / I ) = 0. • If I is a non-zero ideal which it has the a b s o rption p rop ert y, then I is an evolution ideal and has the extension p rop ert y . Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Basic facts ab out evolution algeb ras Abso rption p rop ert y . Prop erties Definition Let I b e an ideal of an evolution algeb ra A . I has the abso rption p rop ert y if xA ⊆ I implies x ∈ I . Lemma • An ideal I of an evolution algeb ra A has the abso rption p rop ert y if and only if ann ( A / I ) = 0 . • If I is a non-zero ideal which it has the a b s o rption p rop ert y, then I is an evolution ideal and has the extension p rop ert y . Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Basic facts ab out evolution algeb ras Abso rption p rop ert y . Prop erties Definition Let I b e an ideal of an evolution algeb ra A . I has the abso rption p rop ert y if xA ⊆ I implies x ∈ I . Lemma • An ideal I of an evolution algeb ra A has the abso rption p rop ert y if and only if ann ( A / I ) = 0. • If I is a non-zero ideal which it has the abso rption p rop ert y , then I is an evolution ideal and has the extension p rop ert y . Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Decomp osition of an evolution algeb ra Simple evolution algeb ras Example The converse is not true in general. Let A b e an evolution algeb ra with natural basis { e i | i ∈ N } and p r o duct given b y: e 2 1 = e 3 + e 5 e 2 3 = e 1 + e 3 + e 5 e 2 5 = e 5 + e 7 e 2 7 = e 3 + e 5 + e 7 . . . e 2 2 = e 4 + e 6 e 2 4 = e 2 + e 4 + e 6 e 2 6 = e 6 + e 8 e 2 8 = e 4 + e 6 + e 8 . . . Then A satisfies the condition s 1, 2 and 3 but A is not simple as  e 2 1  and  e 2 2  a re t w o non-zero p rop er ideals. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Decomp osition of an evolution algeb ra Simple evolution algeb ras Example The converse is not true in general. Let A b e an evolution algeb ra with natural basis { e i | i ∈ N } and p r o duct given b y: e 2 1 = e 3 + e 5 e 2 3 = e 1 + e 3 + e 5 e 2 5 = e 5 + e 7 e 2 7 = e 3 + e 5 + e 7 . . . e 2 2 = e 4 + e 6 e 2 4 = e 2 + e 4 + e 6 e 2 6 = e 6 + e 8 e 2 8 = e 4 + e 6 + e 8 . . . Then A satisfies the condition s 1, 2 and 3 but A is not simple as  e 2 1  and  e 2 2  a re t w o non-zero p rop er ideals. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Decomp osition of an evolution algeb ra Simple evolution algeb ras Example The converse is not true in general. Let A b e an evolution algeb ra with natural basis { e i | i ∈ N } and p ro duct given b y: e 2 1 = e 3 + e 5 e 2 3 = e 1 + e 3 + e 5 e 2 5 = e 5 + e 7 e 2 7 = e 3 + e 5 + e 7 . . . e 2 2 = e 4 + e 6 e 2 4 = e 2 + e 4 + e 6 e 2 6 = e 6 + e 8 e 2 8 = e 4 + e 6 + e 8 . . . Then A satisfies the condition s 1, 2 and 3 but A is not simple as  e 2 1  and  e 2 2  a re t w o non-zero p rop er ideals. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Decomp osition of an evolution algeb ra Simple evolution algeb ras Example The converse is not true in general. Let A b e an evolution algeb ra with natural basis { e i | i ∈ N } and p ro duct given b y: e 2 1 = e 3 + e 5 e 2 3 = e 1 + e 3 + e 5 e 2 5 = e 5 + e 7 e 2 7 = e 3 + e 5 + e 7 . . . e 2 2 = e 4 + e 6 e 2 4 = e 2 + e 4 + e 6 e 2 6 = e 6 + e 8 e 2 8 = e 4 + e 6 + e 8 . . . Then A satisfies the condition s 1, 2 and 3 but A is not simple as  e 2 1  and  e 2 2  a re t w o non-zero p rop er ideals. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Decomp osition of an evolution algeb ra Simple evolution algeb ras Example The converse is not true in general. Let A b e an evolution algeb ra with natural basis { e i | i ∈ N } and p ro duct given b y: e 2 1 = e 3 + e 5 e 2 3 = e 1 + e 3 + e 5 e 2 5 = e 5 + e 7 e 2 7 = e 3 + e 5 + e 7 . . . e 2 2 = e 4 + e 6 e 2 4 = e 2 + e 4 + e 6 e 2 6 = e 6 + e 8 e 2 8 = e 4 + e 6 + e 8 . . . Then A satisfies the conditions 1, 2 and 3 but A is not simple as  e 2 1  and  e 2 2  a re t w o non-zero p rop er ideals. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Decomp osition of an evolution algeb ra Simple evolution algeb ras Example The converse is not true in general. Let A b e an evolution algeb ra with natural basis { e i | i ∈ N } and p ro duct given b y: e 2 1 = e 3 + e 5 e 2 3 = e 1 + e 3 + e 5 e 2 5 = e 5 + e 7 e 2 7 = e 3 + e 5 + e 7 . . . e 2 2 = e 4 + e 6 e 2 4 = e 2 + e 4 + e 6 e 2 6 = e 6 + e 8 e 2 8 = e 4 + e 6 + e 8 . . . Then A satisfies the conditions 1, 2 and 3 but A is not simple as  e 2 1  and  e 2 2  a re t w o non-zero p rop er ideals . Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Decomp osition of an evolution algeb ra Cha racterization fo r finite dimension Co rolla ry If A is a finite dimensional evolution algeb ra and B a natural basis, then A is simple if and only if | M B ( A ) | 6 = 0 and Λ = D ( i ) fo r every i ∈ Λ. Co rolla ry Let A an evolution algeb ra with dim( A )=n and B = { e i | i ∈ Λ } a natural basis of A . Then A is simple if and on ly if | M B ( A ) | 6 = 0 and B cannot b e reo rdered in such a w a y that: W m × m U m × ( n − m ) 0 ( n − m ) × m Y ( n − m ) × ( n − m ) ! fo r some m ∈ N with m < n and matrices W m × m , U m × ( n − m ) and Y ( n − m ) × ( n − m ) . Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Decomp osition of an evolution algeb ra Cha racterization fo r finite dimension Co rolla ry If A is a finite dimensional evolution algeb ra and B a natural basis, then A is simple if and only if | M B ( A ) | 6 = 0 and Λ = D ( i ) fo r every i ∈ Λ. Co rolla ry Let A an evolution algeb ra with dim( A )=n and B = { e i | i ∈ Λ } a natural basis of A . Then A is simple if and on ly if | M B ( A ) | 6 = 0 and B cannot b e reo rdered in such a w a y that: W m × m U m × ( n − m ) 0 ( n − m ) × m Y ( n − m ) × ( n − m ) ! fo r some m ∈ N with m < n and matrices W m × m , U m × ( n − m ) and Y ( n − m ) × ( n − m ) . Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Decomp osition of an evolution algeb ra Cha racterization fo r finite dimension Co rolla ry If A is a finite dimensional evolution algeb ra and B a natural basis, then A is simple if and only if | M B ( A ) | 6 = 0 and Λ = D ( i ) fo r every i ∈ Λ. Co rolla ry Let A an evolution algeb ra with dim( A )=n and B = { e i | i ∈ Λ } a natural basis of A . Then A is simple if and on ly if | M B ( A ) | 6 = 0 and B cannot b e reo rdered in such a w a y that: W m × m U m × ( n − m ) 0 ( n − m ) × m Y ( n − m ) × ( n − m ) ! fo r some m ∈ N with m < n and matrices W m × m , U m × ( n − m ) and Y ( n − m ) × ( n − m ) . Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Decomp osition of an evolution algeb ra Cha racterization fo r finite dimension Co rolla ry If A is a finite dimensional evolution algeb ra and B a natural basis, then A is simple if and only if | M B ( A ) | 6 = 0 and Λ = D ( i ) fo r every i ∈ Λ. Co rolla ry Let A an evolution algeb ra with dim( A )=n and B = { e i | i ∈ Λ } a natural basis of A . Then A is simple if and on ly if | M B ( A ) | 6 = 0 and B cannot b e reo rdered in such a w a y that: W m × m U m × ( n − m ) 0 ( n − m ) × m Y ( n − m ) × ( n − m ) ! fo r some m ∈ N with m < n and matrices W m × m , U m × ( n − m ) and Y ( n − m ) × ( n − m ) . Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Decomp osition of an evolution algeb ra Reducible evolution algeb ras Definition An evolution algeb ra is called reducible if it can b e decomp osed as the direct sum of t w o non-zero evolution subalgeb ras. Otherwise A is called irreducible . Theo rem Let A b e a non-degenerate evolution algeb ra with a natural basis B = { e i | i ∈ Λ } . Assume that A = ⊕ γ ∈ Γ I γ , where each I γ is an ideal of A . Then, there exists a disjoint decomp osition of Λ, sa y Λ = t γ ∈ Γ Λ γ such that I γ = lin { e i | i ∈ Λ γ } . Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Decomp osition of an evolution algeb ra Reducible evolution algeb ras Definition An evolution algeb ra is called reducible if it can b e decomp osed as the direct sum of t w o non-zero evolution subalgeb ras. Otherwise A is called irreducible . Theo rem Let A b e a non-degenerate evolution algeb ra with a natural basis B = { e i | i ∈ Λ } . Assume that A = ⊕ γ ∈ Γ I γ , where each I γ is an ideal of A . Then, there exists a disjoint decomp osition of Λ, sa y Λ = t γ ∈ Γ Λ γ such that I γ = lin { e i | i ∈ Λ γ } . Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Decomp osition of an evolution algeb ra Reducible evolution algeb ras Definition An evolution algeb ra is called reducible if it can b e decomp osed as the direct sum of t w o non-zero evolution subalgeb ras. Otherwise A is called irreducible . Theo rem Let A b e a non-degenerate evolution algeb ra with a natural basis B = { e i | i ∈ Λ } . Assume that A = ⊕ γ ∈ Γ I γ , where each I γ is an ideal of A . Then, there exists a disjoint decomp osition of Λ, sa y Λ = t γ ∈ Γ Λ γ such that I γ = lin { e i | i ∈ Λ γ } . Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Decomp osition of an evolution algeb ra Reducible evolution algeb ras Definition An evolution algeb ra is called reducible if it can b e decomp osed as the direct sum of t w o non-zero evolution subalgeb ras. Otherwise A is called irreducible . Theo rem Let A b e a non-degenerate evolution algeb ra with a natural basis B = { e i | i ∈ Λ } . Assume that A = ⊕ γ ∈ Γ I γ , where each I γ is an ideal of A . Then, there exists a disjoint decomp osition of Λ, sa y Λ = t γ ∈ Γ Λ γ such that I γ = lin { e i | i ∈ Λ γ } . Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Decomp osition of an evolution algeb ra Reducible evolution algeb ras Definition An evolution algeb ra is called reducible if it can b e decomp osed as the direct sum of t w o non-zero evolution subalgeb ras. Otherwise A is called irreducible . Theo rem Let A b e a non-degenerate evolution algeb ra with a natural basis B = { e i | i ∈ Λ } . Assume that A = ⊕ γ ∈ Γ I γ , where each I γ is an ideal of A . Then, there exists a disjoint decomp osition of Λ, sa y Λ = t γ ∈ Γ Λ γ such that I γ = lin { e i | i ∈ Λ γ } . Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Decomp osition of an evolution algeb ra Reducible evolution algeb ras Definition An evolution algeb ra is called reducible if it can b e decomp osed as the direct sum of t w o non-zero evolution subalgeb ras. Otherwise A is called irreducible . Theo rem Let A b e a non-degenerate evolution algeb ra with a natural basis B = { e i | i ∈ Λ } . Assume that A = ⊕ γ ∈ Γ I γ , where each I γ is an ideal of A . Then, there exists a disjoint decomp osition of Λ, sa y Λ = t γ ∈ Γ Λ γ such that I γ = lin { e i | i ∈ Λ γ } . Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Decomp osition of an evolution algeb ra Reducible evolution algeb ras Definition An evolution algeb ra is called reducible if it can b e decomp osed as the direct sum of t w o non-zero evolution subalgeb ras. Otherwise A is called irreducible . Theo rem Let A b e a non-degenerate evolution algeb ra with a natural basis B = { e i | i ∈ Λ } . Assume that A = ⊕ γ ∈ Γ I γ , where each I γ is an ideal of A . Then, there exists a disjoint decomp osition of Λ, sa y Λ = t γ ∈ Γ Λ γ such that I γ = lin { e i | i ∈ Λ γ } . Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Decomp osition of an evolution algeb ra Cha racterizations Rema rk A non-degenerate finite dimensional evolution algeb ra A with natural basis B = { e i | i = 1 ,. .., n } is reducible if and only if there exists B 0 = { e σ ( i ) | i = 1 ,. .., n } with σ ∈ S n such that M B 0 = W m × m 0 ( n − m ) × ( n − m ) 0 ( n − m ) × m Y ( n − m ) × ( n − m ) ! Co rolla ry Let A b e a non-degenerate evolution algeb ra, B = { e i | i ∈ Λ } a natural basis and let E b e its ass o ciated graph. Then A es irreducible if and only if E is a connecte d graph. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Decomp osition of an evolution algeb ra Cha racterizations Rema rk A non-degenerate finite dimensional evolution algeb ra A with natural basis B = { e i | i = 1 ,. .., n } is reducible if and only if there exists B 0 = { e σ ( i ) | i = 1 ,. .., n } with σ ∈ S n such that M B 0 = W m × m 0 ( n − m ) × ( n − m ) 0 ( n − m ) × m Y ( n − m ) × ( n − m ) ! Co rolla ry Let A b e a non-degenerate evolution algeb ra, B = { e i | i ∈ Λ } a natural basis and let E b e its ass o ciated graph. Then A es irreducible if and only if E is a connecte d graph. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Decomp osition of an evolution algeb ra Cha racterizations Rema rk A non-degenerate finite dimensional evolution algeb ra A with natural basis B = { e i | i = 1 ,..., n } is reducible if and only if there exists B 0 = { e σ ( i ) | i = 1 ,. .., n } with σ ∈ S n such that M B 0 = W m × m 0 ( n − m ) × ( n − m ) 0 ( n − m ) × m Y ( n − m ) × ( n − m ) ! Co rolla ry Let A b e a non-degenerate evolution algeb ra, B = { e i | i ∈ Λ } a natural basis and let E b e its ass o ciated graph. Then A es irreducible if and only if E is a connecte d graph. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Decomp osition of an evolution algeb ra Optimal direct-sum decomp osition Definition Let A b e a non-zero evolution algeb ra and assume that A = ⊕ γ ∈ Γ I γ is a direct sum of irreduc ible non-zero ideals, then w e sa y that A = ⊕ γ ∈ Γ I γ is an optimal direct-sum decomp osition of A . Theo rem Let A b e a non-degenerate evolution algeb ra. Then A admits an optimal direct-sum decomp osition. Mo reover, it is unique. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Decomp osition of an evolution algeb ra Optimal direct-sum decomp osition Definition Let A b e a non-zero evolution algeb ra and assume that A = ⊕ γ ∈ Γ I γ is a direct sum of irreduc ible non-zero ideals, then w e sa y that A = ⊕ γ ∈ Γ I γ is an optimal direct-sum decomp osition of A . Theo rem Let A b e a non-degenerate evolution algeb ra. Then A admits an optimal direct-sum decomp osition. Mo reover, it is unique. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Decomp osition of an evolution algeb ra Optimal direct-sum decomp osition Definition Let A b e a non-zero evolution algeb ra and assume that A = ⊕ γ ∈ Γ I γ is a direct sum of irreducible non-zero ideals, then w e sa y that A = ⊕ γ ∈ Γ I γ is an optimal direct-sum decomp osition of A . Theo rem Let A b e a non-degenerate evolution algeb ra. Then A admits an optimal direct-sum decomp osition. Mo reover, it is unique. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Decomp osition of an evolution algeb ra Optimal direct-sum decomp osition Definition Let A b e a non-zero evolution algeb ra and assume that A = ⊕ γ ∈ Γ I γ is a direct sum of irreducible non-zero ideals, then w e sa y that A = ⊕ γ ∈ Γ I γ is an optimal direct-sum decomp osition of A . Theo rem Let A b e a non-degenerate evolution algeb ra. Then A admits an optimal direct-sum decomp osition. Mo reover, it is unique. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Decomp osition of an evolution algeb ra Optimal direct-sum decomp osition Definition Let A b e a non-zero evolution algeb ra and assume that A = ⊕ γ ∈ Γ I γ is a direct sum of irreducible non-zero ideals, then w e sa y that A = ⊕ γ ∈ Γ I γ is an optimal direct-sum decomp osition of A . Theo rem Let A b e a non-degenerate evolution algeb ra. Then A admits an optimal direct-sum decomp osition. Mo reover, it is unique. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Decomp osition of an evolution algeb ra Optimal direct-sum decomp osition Definition Let A b e a non-zero evolution algeb ra and assume that A = ⊕ γ ∈ Γ I γ is a direct sum of irreducible non-zero ideals, then w e sa y that A = ⊕ γ ∈ Γ I γ is an optimal direct-sum decomp osition of A . Theo rem Let A b e a non-degenerate evolution algeb ra. Then A admits an optimal direct-sum decomp osition. Mo reover, it is unique. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Decomp osition of an evolution algeb ra Optimal direct-sum decomp osition Definition Let A b e a non-zero evolution algeb ra and assume that A = ⊕ γ ∈ Γ I γ is a direct sum of irreducible non-zero ideals, then w e sa y that A = ⊕ γ ∈ Γ I γ is an optimal direct-sum decomp osition of A . Theo rem Let A b e a non-degenerate evolution algeb ra. Then A admits an optimal direct-sum decomp osition . Mo reover, it is unique. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Decomp osition of an evolution algeb ra Optimal direct-sum decomp osition Definition Let A b e a non-zero evolution algeb ra and assume that A = ⊕ γ ∈ Γ I γ is a direct sum of irreducible non-zero ideals, then w e sa y that A = ⊕ γ ∈ Γ I γ is an optimal direct-sum decomp osition of A . Theo rem Let A b e a non-degenerate evolution algeb ra. Then A admits an optimal direct-sum decomp osition . Mo reover, it is unique . Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Decomp osition of an evolution algeb ra The fragmentation p ro cess Definition Let Λ b e a finite set and let Υ 1 ,. .., Υ n b e non-empt y subsets of Λ such that Λ = ∪ n i =1 Υ i . W e sa y that Λ = ∪ n i =1 Υ i is a fragmentable union if there e xists disjoint non-empt y subsets Λ 1 , Λ 2 of Λ satisfying Λ = ∪ n i =1 Υ i = Λ 1 ∪ Λ 2 , and such that fo r e very i = 1 ,. .., n , either Υ i ⊆ Λ 1 o rΥ i ⊆ Λ 2 . Definition A fragmentation of a fragmentable union Λ = ∪ n i =1 Υ i is a union Λ = ∪ k i =1 Λ i such that: 1 If i ∈ { 1 ,. .., k } then Λ i = ∪ j ∈ S i Υ j fo r S i a non-empt y subset of { 1 ,. .., n } . 2 Λ i ∩ Λ j = ∅ , fo r ever y i , j ∈ { 1 ,. .., k } , with i 6 = j . If fo r every i ∈ { 1 ,. .., k } the index s e t Λ i = ∪ j ∈ S i Υ j is not fragmentable then w e sa y that Λ = ∪ k i =1 Λ i is an optimal fragmentation. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Decomp osition of an evolution algeb ra The fragmentation p ro cess Definition Let Λ b e a finite set and let Υ 1 ,. .., Υ n b e non-empt y subsets of Λ such that Λ = ∪ n i =1 Υ i . W e sa y that Λ = ∪ n i =1 Υ i is a fragmentable union if there e xists disjoint non-empt y subsets Λ 1 , Λ 2 of Λ satisfying Λ = ∪ n i =1 Υ i = Λ 1 ∪ Λ 2 , and such that fo r e very i = 1 ,. .., n , either Υ i ⊆ Λ 1 o rΥ i ⊆ Λ 2 . Definition A fragmentation of a fragmentable union Λ = ∪ n i =1 Υ i is a union Λ = ∪ k i =1 Λ i such that: 1 If i ∈ { 1 ,. .., k } then Λ i = ∪ j ∈ S i Υ j fo r S i a non-empt y subset of { 1 ,. .., n } . 2 Λ i ∩ Λ j = ∅ , fo r ever y i , j ∈ { 1 ,. .., k } , with i 6 = j . If fo r every i ∈ { 1 ,. .., k } the index s e t Λ i = ∪ j ∈ S i Υ j is not fragmentable then w e sa y that Λ = ∪ k i =1 Λ i is an optimal fragmentation. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Decomp osition of an evolution algeb ra The fragmentation p ro cess Definition Let Λ b e a finite set and let Υ 1 ,..., Υ n be non-empty subsets of Λ such that Λ = ∪ n i =1 Υ i . We sa y that Λ = ∪ n i =1 Υ i is a fragmentable union if there exists disjoint non-empt y subsets Λ 1 , Λ 2 of Λ satisfying Λ = ∪ n i =1 Υ i = Λ 1 ∪ Λ 2 , and such that fo r every i = 1 ,..., n , either Υ i ⊆ Λ 1 orΥ i ⊆ Λ 2 . Definition A fragmentation of a fragmentable union Λ = ∪ n i =1 Υ i is a union Λ = ∪ k i =1 Λ i such that: 1 If i ∈ { 1 ,. .., k } then Λ i = ∪ j ∈ S i Υ j fo r S i a non-empt y subset of { 1 ,. .., n } . 2 Λ i ∩ Λ j = ∅ , fo r ever y i , j ∈ { 1 ,. .., k } , with i 6 = j . If fo r every i ∈ { 1 ,. .., k } the index s e t Λ i = ∪ j ∈ S i Υ j is not fragmentable then w e sa y that Λ = ∪ k i =1 Λ i is an optimal fragmentation. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Decomp osition of an evolution algeb ra The fragmentation p ro cess Definition Let Λ b e a finite set and let Υ 1 ,..., Υ n be non-empty subsets of Λ such that Λ = ∪ n i =1 Υ i . We sa y that Λ = ∪ n i =1 Υ i is a fragmentable union if there exists disjoint non-empt y subsets Λ 1 , Λ 2 of Λ satisfying Λ = ∪ n i =1 Υ i = Λ 1 ∪ Λ 2 , and such that fo r every i = 1 ,..., n , either Υ i ⊆ Λ 1 orΥ i ⊆ Λ 2 . Definition A fragmentation of a fragmentable union Λ = ∪ n i =1 Υ i is a union Λ = ∪ k i =1 Λ i such that: 1 If i ∈ { 1 ,. .., k } then Λ i = ∪ j ∈ S i Υ j fo r S i a non-empt y subset of { 1 ,. .., n } . 2 Λ i ∩ Λ j = ∅ , fo r ever y i , j ∈ { 1 ,. .., k } , with i 6 = j . If fo r every i ∈ { 1 ,. .., k } the index s e t Λ i = ∪ j ∈ S i Υ j is not fragmentable then w e sa y that Λ = ∪ k i =1 Λ i is an optimal fragmentation. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Decomp osition of an evolution algeb ra The fragmentation p ro cess Definition Let Λ b e a finite set and let Υ 1 ,..., Υ n be non-empty subsets of Λ such that Λ = ∪ n i =1 Υ i . We sa y that Λ = ∪ n i =1 Υ i is a fragmentable union if there exists disjoint non-empt y subsets Λ 1 , Λ 2 of Λ satisfying Λ = ∪ n i =1 Υ i = Λ 1 ∪ Λ 2 , and such that fo r every i = 1 ,..., n , either Υ i ⊆ Λ 1 orΥ i ⊆ Λ 2 . Definition A fragmentation of a fragmentable union Λ = ∪ n i =1 Υ i is a union Λ = ∪ k i =1 Λ i such that: 1 If i ∈ { 1 ,..., k } then Λ i = ∪ j ∈ S i Υ j fo r S i a non-empty subset of { 1 ,..., n } . 2 Λ i ∩ Λ j = ∅ , fo r ever y i , j ∈ { 1 ,. .., k } , with i 6 = j . If fo r every i ∈ { 1 ,. .., k } the index s e t Λ i = ∪ j ∈ S i Υ j is not fragmentable then w e sa y that Λ = ∪ k i =1 Λ i is an optimal fragmentation. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Decomp osition of an evolution algeb ra The fragmentation p ro cess Definition Let Λ b e a finite set and let Υ 1 ,..., Υ n be non-empty subsets of Λ such that Λ = ∪ n i =1 Υ i . We sa y that Λ = ∪ n i =1 Υ i is a fragmentable union if there exists disjoint non-empt y subsets Λ 1 , Λ 2 of Λ satisfying Λ = ∪ n i =1 Υ i = Λ 1 ∪ Λ 2 , and such that fo r every i = 1 ,..., n , either Υ i ⊆ Λ 1 orΥ i ⊆ Λ 2 . Definition A fragmentation of a fragmentable union Λ = ∪ n i =1 Υ i is a union Λ = ∪ k i =1 Λ i such that: 1 If i ∈ { 1 ,..., k } then Λ i = ∪ j ∈ S i Υ j fo r S i a non-empty subset of { 1 ,..., n } . 2 Λ i ∩ Λ j = ∅ , fo r every i , j ∈ { 1 ,..., k } , with i 6 = j . If fo r every i ∈ { 1 ,. .., k } the index s e t Λ i = ∪ j ∈ S i Υ j is not fragmentable then w e sa y that Λ = ∪ k i =1 Λ i is an optimal fragmentation. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Decomp osition of an evolution algeb ra The fragmentation p ro cess Definition Let Λ b e a finite set and let Υ 1 ,..., Υ n be non-empty subsets of Λ such that Λ = ∪ n i =1 Υ i . We sa y that Λ = ∪ n i =1 Υ i is a fragmentable union if there exists disjoint non-empt y subsets Λ 1 , Λ 2 of Λ satisfying Λ = ∪ n i =1 Υ i = Λ 1 ∪ Λ 2 , and such that fo r every i = 1 ,..., n , either Υ i ⊆ Λ 1 orΥ i ⊆ Λ 2 . Definition A fragmentation of a fragmentable union Λ = ∪ n i =1 Υ i is a union Λ = ∪ k i =1 Λ i such that: 1 If i ∈ { 1 ,..., k } then Λ i = ∪ j ∈ S i Υ j fo r S i a non-empty subset of { 1 ,..., n } . 2 Λ i ∩ Λ j = ∅ , fo r every i , j ∈ { 1 ,..., k } , with i 6 = j . If fo r every i ∈ { 1 ,..., k } the index set Λ i = ∪ j ∈ S i Υ j is not fragmentable then w e sa y that Λ = ∪ k i =1 Λ i is an optimal fragmentation. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Decomp osition of an evolution algeb ra The fragmentation p ro cess Definition Let Λ b e a finite set and let Υ 1 ,..., Υ n be non-empty subsets of Λ such that Λ = ∪ n i =1 Υ i . We sa y that Λ = ∪ n i =1 Υ i is a fragmentable union if there exists disjoint non-empt y subsets Λ 1 , Λ 2 of Λ satisfying Λ = ∪ n i =1 Υ i = Λ 1 ∪ Λ 2 , and such that fo r every i = 1 ,..., n , either Υ i ⊆ Λ 1 orΥ i ⊆ Λ 2 . Definition A fragmentation of a fragmentable union Λ = ∪ n i =1 Υ i is a union Λ = ∪ k i =1 Λ i such that: 1 If i ∈ { 1 ,..., k } then Λ i = ∪ j ∈ S i Υ j fo r S i a non-empty subset of { 1 ,..., n } . 2 Λ i ∩ Λ j = ∅ , fo r every i , j ∈ { 1 ,..., k } , with i 6 = j . If fo r every i ∈ { 1 ,..., k } the index set Λ i = ∪ j ∈ S i Υ j is not fragmentable then w e say that Λ = ∪ k i =1 Λ i is an optimal fragmentation . Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Decomp osition of an evolution algeb ra The optimal direct-sum decomp osition of A Theo rem (The fragmentation p ro cess) Let A b e a finite-dimensional evolution algeb ra with natural basis B = { e i | i ∈ Λ } . Let { C 1 ,. .., C k } b e the set of p rincipal cycles of Λ , { i 1 ,. .., i m } the set of all chain-sta rt indices of Λ and consider the decomp osition ( † ) Λ = Λ( C 1 ) ∪ · · · ∪ Λ( C k ) ∪ Λ( i 1 ) ∪ · · · ∪ Λ( i m ) . where Λ( S ) := S ∪ i ∈ S D ( i ). Let Λ = t γ ∈ Γ Λ γ b e the optimal fragmentation of ( † ) and decomp ose B = t γ ∈ Γ B γ , where B γ = { e i | i ∈ Λ γ } . Then A = ⊕ γ ∈ Γ I γ , fo r I γ = lin B γ , which is an evolution ideal of A . Mo reover, if A is non-degenerate, then A = ⊕ γ ∈ Γ I γ is the optimal direct-sum decomp osition of A . Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Decomp osition of an evolution algeb ra The optimal direct-sum decomp osition of A Theo rem (The fragmentation p ro cess) Let A b e a finite-dimensional evolution algeb ra with natural basis B = { e i | i ∈ Λ } . Let { C 1 ,. .., C k } b e the set of p rincipal cycles of Λ , { i 1 ,. .., i m } the set of all chain-sta rt indices of Λ and consider the decomp osition ( † ) Λ = Λ( C 1 ) ∪ · · · ∪ Λ( C k ) ∪ Λ( i 1 ) ∪ · · · ∪ Λ( i m ) . where Λ( S ) := S ∪ i ∈ S D ( i ). Let Λ = t γ ∈ Γ Λ γ b e the optimal fragmentation of ( † ) and decomp ose B = t γ ∈ Γ B γ , where B γ = { e i | i ∈ Λ γ } . Then A = ⊕ γ ∈ Γ I γ , fo r I γ = lin B γ , which is an evolution ideal of A . Mo reover, if A is non-degenerate, then A = ⊕ γ ∈ Γ I γ is the optimal direct-sum decomp osition of A . Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Decomp osition of an evolution algeb ra The optimal direct-sum decomp osition of A Theo rem (The fragmentation p ro cess) Let A b e a finite-dimensional evolution algeb ra with natural basis B = { e i | i ∈ Λ } . Let { C 1 ,. .., C k } b e the set of p rincipal cycles of Λ , { i 1 ,. .., i m } the set of all chain-sta rt indices of Λ and consider the decomp osition ( † ) Λ = Λ( C 1 ) ∪ · · · ∪ Λ( C k ) ∪ Λ( i 1 ) ∪ · · · ∪ Λ( i m ) . where Λ( S ) := S ∪ i ∈ S D ( i ). Let Λ = t γ ∈ Γ Λ γ b e the optimal fragmentation of ( † ) and decomp ose B = t γ ∈ Γ B γ , where B γ = { e i | i ∈ Λ γ } . Then A = ⊕ γ ∈ Γ I γ , fo r I γ = lin B γ , which is an evolution ideal of A . Mo reover, if A is non-degenerate, then A = ⊕ γ ∈ Γ I γ is the optimal direct-sum decomp osition of A . Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Decomp osition of an evolution algeb ra The optimal direct-sum decomp osition of A Theo rem (The fragmentation p ro cess) Let A b e a finite-dimensional evolution algeb ra with natural basis B = { e i | i ∈ Λ } . Let { C 1 ,..., C k } b e the set of p rincipal cycles of Λ , { i 1 ,..., i m } the set of all chain-sta rt indices of Λ and consider the decomp osition ( † ) Λ = Λ( C 1 ) ∪ · · · ∪ Λ( C k ) ∪ Λ( i 1 ) ∪ · · · ∪ Λ( i m ) . where Λ( S ) := S ∪ i ∈ S D ( i ). Let Λ = t γ ∈ Γ Λ γ b e the optimal fragmentation of ( † ) and decomp ose B = t γ ∈ Γ B γ , where B γ = { e i | i ∈ Λ γ } . Then A = ⊕ γ ∈ Γ I γ , fo r I γ = lin B γ , which is an evolution ideal of A . Mo reover, if A is non-degenerate, then A = ⊕ γ ∈ Γ I γ is the optimal direct-sum decomp osition of A . Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Decomp osition of an evolution algeb ra Optimal Fragmentation PROGRAM 1 Program A In[1]:= P = 00001001 00100010 00000100 00100000 00001000 00000000 00000001 10000011 ; D 1 [ i _ , P _] : = Select [ Table [ j, { j, Length [ P ]}] , P [[# , i ] ]≠ 0 & ] ; D 1 [ 5, P ] D m _ [ i _ , P _] : = Module [{ a, k, s } , a = {} ;s = Length [ D m -− 1 [ i, P ]] ; If [ m == 1, D 1 [ i, P ] , Union [ Flatten [ Table [ D 1 [ D m -− 1 [ i, P ][[ t ]] , P ] , { t, Length [ D m -− 1 [ i, P ]]}]]]]] ; CycleQ [ P _] : = Module [{ n, x } ,n = Length [ P ] ; x = Union [ Flatten [ Table [ Diagonal [ MatrixPower [ P, i ]] , { i, 1, n }]]] ; MemberQ [ x, 1 ]] ; CycleQ [ P ] DYesCycle [ i _ , P _] : = Module [{ b, j } , b = {} ; For [ j = 1 ,j ≤ Length [ P ] , j ++ , AppendTo [ b, D j [ i, P ]]] ; Apply [ Union , b ]] ; DNotCycle [ i _ , P _] : = Module [{ t, b } , b = { D 1 [ i, P ]} ; For [ t = 1, D t [ i, P ]≠ D t + 1 [ i, P ] , t ++ , AppendTo [ b, D t + 1 [ i, P ]] ] ; Apply [ Union , b ]] ; DP [ i _ , P _] : = If [ CycleQ [ P ] , DYesCycle [ i, P ] , DNotCycle [ i, P ]] Out[3]= { 1, 5 } Out[6]= True In[10]:= DP [ 5, P ] Out[10]= { 1, 2, 5, 7, 8 } Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Decomp osition of an evolution algeb ra Program B In[11]:= CyclicQ [ i _ , P _] : = If [ MemberQ [ DP [ i ] , i ] , Print [ i " is a cyclic index " ] , Print [ i " is not a cyclic index " ]] ; CycleAssociated [ i _ , P _] : = Select [ Table [ j, { j, Length [ P ]}] , MemberQ [ DP [ i, P ] , #] && MemberQ [ DP [# , P ] , i ] & ] ; Ascendents [ i _ , P _] : = Module [{ j, b } , b = {} ; For [ j = 1 ,j ≤ Length [ P ] , j ++ , If [ MemberQ [ DP [ j, P ] , i ] , AppendTo [ b, j ]]] ; b ] ; Subset [ A _ , B _] : =  Union [ A, B ] ⩵ Union [ B ] ; PrincipalCycleQ [ i _ , P _] : = If [ Subset [ Ascendents [ i, P ] , CycleAssociated [ i, P ]] , Print [ i " is a principal cyclic -− index " ] , Print [ i " is not a principal cyclic -− index " ]] ; ElementsNotNoneRow [ P _] : = Module [{ j } , Select [ Table [ j, { j, Length [ P ]}] , P [[#]] ⩵ 0 P [[ 1 ]] & ]] ; (*⋆ La función incad me devuelve los índices i tales que la fila i es nula *⋆) ChainStartQ [ i _ , P _] : = If [ MemberQ [ ElementsNotNoneRow [ P ] , i ] , Print [ i " is a chain -− start index " ] , Print [ i " is not a chain -− start index " ]] ; CycleAssociated [ 5, P ] Ascendents [ 5, P ] Out[18]= { 5 } Out[19]= { 5 } Program C In[20]:= LambdaPrincipalCycle [ P _] : = Module [{ j, b } , b = {} ; For [ j = 1 ,j ≤ Length [ P ] , j ++ , If [ Subset [ Ascendents [ j, P ] , CycleAssociated [ j, P ]] , AppendTo [ b, DP [ j, P ]]]] ; b ] ; Λ[ i _ , P _] : = Union [{ i } , DP [ i, P ]] ; LambdaChainStart [ P _] : = Table [Λ[ ElementsNotNoneRow [ P ][[ i ]] , P ] , { i, Length [ ElementsNotNoneRow [ P ]]}] CanonicalDecomposition [ P _] : = Join [ LambdaChainStart [ P ] , LambdaPrincipalCycle [ P ]] ; CanonicalDecomposition [ P ] Out[24]= {{ 2, 3, 4, 6 } , { 1, 2, 5, 7, 8 } , { 2, 3, 4 }} PROGRAM 2 2 programa dimension 8.nb Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Decomp osition of an evolution algeb ra In[25]:= f [ i _ , j _ , P _] : = If [ i ⩵ j, 0, If [ Intersection [ Part [ CanonicalDecomposition [ P ] , i ] , Part [ CanonicalDecomposition [ P ] , j ]] ≠ {} , 1, 0 ]] ; Matr [ P _] : = Table [ f [ i, j, P ] , { i, Length [ CanonicalDecomposition [ P ]]} , { j, Length [ CanonicalDecomposition [ P ]]}] ; AdjacencyGraph [ Matr [ P ] , VertexLabels → " Name " ] OptimalFragmentation [ P _] : = ConnectedComponents [ AdjacencyGraph [ Matr [ P ] , VertexLabels → " Name " ]] OptimalFragmentation [ P ] Out[27]= Out[29]= ( 1 2 3 ) programa dimension 8.nb 3 Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Classification t w o-dimensional evolution algeb ras Outline 1 Intro duction 2 Basic facts ab out evolution algeb ras Evolution algeb ras Pro duct and Change of basis Subalgeb ras and ideals Non-degenerate evolution algeb ras The graph asso ciated to an evolution algeb ra 3 Decomp osition of an evolution algeb ra Ideals generated b y one element Simple evolution algeb ras Reducible evolution algeb ras The optimal direct-sum decomp osition of an evolution algeb ra 4 Classification t w o-dimensional evolution algeb ras 5 Classification of three-dimensional evolution algeb ras 6 F urther w o rk Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Classification t w o-dimensional evolution algeb ras Classification of 2-dimensional complex evolution algeb ras Theo rem (Casas J.M., Ladra M., Omirov B.A., and Rozik ov U.A., 2014) Any 2-dimensional complex evolution algeb ra E is isomo rphic to one of the follo wing pa ir wis e non isomo rphic algeb ras: 1 dim E 2 =1 • E 1 : e 1 e 1 = e 1 , • E 2 : e 1 e 1 = e 1 , e 2 e 2 = e 1 , • E 3 : e 1 e 1 = e 1 + e 2 , e 2 e 2 = − e 1 − e 2 , • E 4 : e 1 e 1 = e 2 . 2 dim E 2 =2 • E 5 : e 1 e 1 = e 1 + a 2 e 2 , e 2 e 2 = a 3 e 1 + e 2 , 1 − a 2 a 3 6 = 0, where E 5 ( a 2 , a 3 ) ∼ = E 0 5 ( a 3 , a 2 ), • E 6 : e 1 e 1 = e 2 , e 2 e 2 = e 1 + a 4 e 2 , a 4 6 = 0, where E 6 ( a 4 ) ∼ = E 6 ( a 0 4 ) ⇔ a 0 4 a 4 = cos 2 π k 3 + ı sin 2 π k 3 fo r some k = 0 , 1 , 2. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Classification t w o-dimensional evolution algeb ras Classification of 2-dimensional complex evolution algeb ras Theo rem (Casas J.M., Ladra M., Omirov B.A., and Rozik ov U.A., 2014) Any 2-dimensional complex evolution algeb ra E is isomo rphic to one of the follo wing pa ir wis e non isomo rphic algeb ras: 1 dim E 2 =1 • E 1 : e 1 e 1 = e 1 , • E 2 : e 1 e 1 = e 1 , e 2 e 2 = e 1 , • E 3 : e 1 e 1 = e 1 + e 2 , e 2 e 2 = − e 1 − e 2 , • E 4 : e 1 e 1 = e 2 . 2 dim E 2 =2 • E 5 : e 1 e 1 = e 1 + a 2 e 2 , e 2 e 2 = a 3 e 1 + e 2 , 1 − a 2 a 3 6 = 0, where E 5 ( a 2 , a 3 ) ∼ = E 0 5 ( a 3 , a 2 ), • E 6 : e 1 e 1 = e 2 , e 2 e 2 = e 1 + a 4 e 2 , a 4 6 = 0, where E 6 ( a 4 ) ∼ = E 6 ( a 0 4 ) ⇔ a 0 4 a 4 = cos 2 π k 3 + ı sin 2 π k 3 fo r some k = 0 , 1 , 2. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Classification t w o-dimensional evolution algeb ras Classification of 2-dimensional complex evolution algeb ras Theo rem (Casas J.M., Ladra M., Omirov B.A., and Rozik ov U.A., 2014) Any 2-dimensional complex evolution algeb ra E is isomo rphic to one of the follo wing pairwise non isomo rphic algeb ras: 1 dim E 2 =1 • E 1 : e 1 e 1 = e 1 , • E 2 : e 1 e 1 = e 1 , e 2 e 2 = e 1 , • E 3 : e 1 e 1 = e 1 + e 2 , e 2 e 2 = − e 1 − e 2 , • E 4 : e 1 e 1 = e 2 . 2 dim E 2 =2 • E 5 : e 1 e 1 = e 1 + a 2 e 2 , e 2 e 2 = a 3 e 1 + e 2 , 1 − a 2 a 3 6 = 0, where E 5 ( a 2 , a 3 ) ∼ = E 0 5 ( a 3 , a 2 ), • E 6 : e 1 e 1 = e 2 , e 2 e 2 = e 1 + a 4 e 2 , a 4 6 = 0, where E 6 ( a 4 ) ∼ = E 6 ( a 0 4 ) ⇔ a 0 4 a 4 = cos 2 π k 3 + ı sin 2 π k 3 fo r some k = 0 , 1 , 2. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Classification t w o-dimensional evolution algeb ras Classification of 2-dimensional complex evolution algeb ras Theo rem (Casas J.M., Ladra M., Omirov B.A., and Rozik ov U.A., 2014) Any 2-dimensional complex evolution algeb ra E is isomo rphic to one of the follo wing pairwise non isomo rphic algeb ras: 1 dim E 2 =1 • E 1 : e 1 e 1 = e 1 , • E 2 : e 1 e 1 = e 1 , e 2 e 2 = e 1 , • E 3 : e 1 e 1 = e 1 + e 2 , e 2 e 2 = − e 1 − e 2 , • E 4 : e 1 e 1 = e 2 . 2 dim E 2 =2 • E 5 : e 1 e 1 = e 1 + a 2 e 2 , e 2 e 2 = a 3 e 1 + e 2 , 1 − a 2 a 3 6 = 0, where E 5 ( a 2 , a 3 ) ∼ = E 0 5 ( a 3 , a 2 ), • E 6 : e 1 e 1 = e 2 , e 2 e 2 = e 1 + a 4 e 2 , a 4 6 = 0, where E 6 ( a 4 ) ∼ = E 6 ( a 0 4 ) ⇔ a 0 4 a 4 = cos 2 π k 3 + ı sin 2 π k 3 fo r some k = 0 , 1 , 2. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Classification t w o-dimensional evolution algeb ras Classification of 2-dimensional complex evolution algeb ras Theo rem (Casas J.M., Ladra M., Omirov B.A., and Rozik ov U.A., 2014) Any 2-dimensional complex evolution algeb ra E is isomo rphic to one of the follo wing pairwise non isomo rphic algeb ras: 1 dim E 2 =1 • E 1 : e 1 e 1 = e 1 , • E 2 : e 1 e 1 = e 1 , e 2 e 2 = e 1 , • E 3 : e 1 e 1 = e 1 + e 2 , e 2 e 2 = − e 1 − e 2 , • E 4 : e 1 e 1 = e 2 . 2 dim E 2 =2 • E 5 : e 1 e 1 = e 1 + a 2 e 2 , e 2 e 2 = a 3 e 1 + e 2 , 1 − a 2 a 3 6 = 0, where E 5 ( a 2 , a 3 ) ∼ = E 0 5 ( a 3 , a 2 ), • E 6 : e 1 e 1 = e 2 , e 2 e 2 = e 1 + a 4 e 2 , a 4 6 = 0, where E 6 ( a 4 ) ∼ = E 6 ( a 0 4 ) ⇔ a 0 4 a 4 = cos 2 π k 3 + ı sin 2 π k 3 fo r some k = 0 , 1 , 2. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Classification t w o-dimensional evolution algeb ras Classification of 2-dimensional evolution algeb ras Theo rem Let A b e a t w o-dimensional evolution algeb ra over a field K where fo r ever y k ∈ K the p olynomial x n − k has a ro ot whenever n = 2 , 3. If dim ( A 2 ) = 0 then M B = 0 fo r any natural basis B of A . If dim ( A 2 ) = 1 then M B is one of the follo win g four matrices: 1 M B = 1 0 0 0 ! , 2 M B = 1 − 1 1 − 1 ! , 3 M B = 1 1 0 0 ! , 4 M B = 0 1 0 0 ! . They a r e mutually non-isomo rphic. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Classification t w o-dimensional evolution algeb ras Classification of 2-dimensional evolution algeb ras Theo rem If dim ( A 2 ) = 2 then M B is one of the follo wing three types of matrices: 5 M B ( α , β ) = 1 α β 1 ! fo r some α , β ∈ K × and 1 − α β 6 = 0. 6 M B ( α ) = 1 α 0 1 ! fo r some α ∈ K × . 7 M B = 1 0 0 1 ! 8 M B = 0 1 1 0 ! , 9 M B ( γ ) = 0 1 1 γ ! fo r some γ ∈ K × . They a re mutually non-isomo rphic excep t in the case { M B ( γ ) | γ ∈ K } when γ γ 0 is a 3 rd ro ot of unit y . Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Classification t w o-dimensional evolution algeb ras Classification of 2-dimensional evolution algeb ras Theo rem If dim ( A 2 ) = 2 then M B is one of the follo wing three types of matrices: 5 M B ( α, β ) = 1 α β 1 ! fo r some α , β ∈ K × and 1 − αβ 6 = 0. 6 M B ( α ) = 1 α 0 1 ! for some α ∈ K × . 7 M B = 1 0 0 1 ! 8 M B = 0 1 1 0 ! , 9 M B ( γ ) = 0 1 1 γ ! fo r some γ ∈ K × . They a re mutually non-isomo rphic excep t in the case { M B ( γ ) | γ ∈ K } when γ γ 0 is a 3 rd ro ot of unit y . Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Classification t w o-dimensional evolution algeb ras Classification of 2-dimensional evolution algeb ras Theo rem If dim ( A 2 ) = 2 then M B is one of the follo wing three types of matrices: 5 M B ( α, β ) = 1 α β 1 ! fo r some α , β ∈ K × and 1 − αβ 6 = 0. 6 M B ( α ) = 1 α 0 1 ! for some α ∈ K × . 7 M B = 1 0 0 1 ! 8 M B = 0 1 1 0 ! , 9 M B ( γ ) = 0 1 1 γ ! fo r some γ ∈ K × . They a re mutually non-isomorphic except in the case { M B ( γ ) | γ ∈ K } when γ γ 0 is a 3 rd ro ot of unit y . Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Classification t w o-dimensional evolution algeb ras dim ( A 2 )=1 Fix a t w o-dimensional evolution algeb ra A and a natural basis B = { e 1 , e 2 } . Let M B = ω 1 ω 3 ω 2 ω 4 ! • Supp ose that { e 2 1 } is a basis of A 2 . Sinc e e 2 2 ∈ A 2 , there exists c 1 ∈ K such that e 2 2 = c 1 e 2 1 = c 1 ( ω 1 e 1 + ω 2 e 2 ). • Extend the basis { e 2 1 } of A 2 to a new basis B 0 = { e 0 1 , e 0 2 } of A ( B 0 is not necessa rily a n a tur a l basis) with ch a nge of basis matrix P B 0 B P B 0 B = ω 1 p 1 ω 2 p 2 ! . Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Classification t w o-dimensional evolution algeb ras dim ( A 2 )=1 Fix a t w o-dimensional evolution algeb ra A and a natural basis B = { e 1 , e 2 } . Let M B = ω 1 ω 3 ω 2 ω 4 ! • Supp ose that { e 2 1 } is a basis of A 2 . Sinc e e 2 2 ∈ A 2 , there exists c 1 ∈ K such that e 2 2 = c 1 e 2 1 = c 1 ( ω 1 e 1 + ω 2 e 2 ). • Extend the basis { e 2 1 } of A 2 to a new basis B 0 = { e 0 1 , e 0 2 } of A ( B 0 is not necessa rily a n a tur a l basis) with ch a nge of basis matrix P B 0 B P B 0 B = ω 1 p 1 ω 2 p 2 ! . Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Classification t w o-dimensional evolution algeb ras dim ( A 2 )=1 Fix a t w o-dimensional evolution algeb ra A and a natural basis B = { e 1 , e 2 } . Let M B = ω 1 ω 3 ω 2 ω 4 ! • Supp ose that { e 2 1 } is a basis of A 2 . Since e 2 2 ∈ A 2 , there exists c 1 ∈ K such that e 2 2 = c 1 e 2 1 = c 1 ( ω 1 e 1 + ω 2 e 2 ). • Extend the basis { e 2 1 } of A 2 to a new basis B 0 = { e 0 1 , e 0 2 } of A ( B 0 is not necessa rily a n a tur a l basis) with ch a nge of basis matrix P B 0 B P B 0 B = ω 1 p 1 ω 2 p 2 ! . Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Classification t w o-dimensional evolution algeb ras dim ( A 2 )=1 Fix a t w o-dimensional evolution algeb ra A and a natural basis B = { e 1 , e 2 } . Let M B = ω 1 ω 3 ω 2 ω 4 ! • Supp ose that { e 2 1 } is a basis of A 2 . Since e 2 2 ∈ A 2 , there exists c 1 ∈ K such that e 2 2 = c 1 e 2 1 = c 1 ( ω 1 e 1 + ω 2 e 2 ). • Extend the basis { e 2 1 } of A 2 to a new basis B 0 = { e 0 1 , e 0 2 } of A ( B 0 is not necessa rily a natural basis) with change of basis matrix P B 0 B P B 0 B = ω 1 p 1 ω 2 p 2 ! . Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Classification t w o-dimensional evolution algeb ras dim ( A 2 )=1 • When is B 0 a natural basis? • ( e 2 1 ) 2 = ω 2 1 e 2 1 + ω 2 2 e 2 2 = ω 2 1 e 2 1 + ω 2 2 ( c 2 e 2 1 )=( ω 2 1 + ω 2 2 c 2 ) e 2 1 . • e 0 2 2 = p 2 1 e 2 1 + p 2 2 e 2 2 = ( p 2 1 + p 2 2 c 2 ) e 2 1 . • e 2 1 e 0 2 = ( ω 1 e 1 + ω 2 e 2 )( p 1 e 1 + p 2 e 2 )=( ω 1 p 1 + ω 2 p 2 c 2 ) e 2 1 = 0 . • Distinguish differe nt cases dep en ding on ω 1 p 1 + ω 2 p 2 c 2 is zero o r not. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Classification t w o-dimensional evolution algeb ras dim ( A 2 )=1 • When is B 0 a natural basis? • ( e 2 1 ) 2 = ω 2 1 e 2 1 + ω 2 2 e 2 2 = ω 2 1 e 2 1 + ω 2 2 ( c 2 e 2 1 )=( ω 2 1 + ω 2 2 c 2 ) e 2 1 . • e 0 2 2 = p 2 1 e 2 1 + p 2 2 e 2 2 = ( p 2 1 + p 2 2 c 2 ) e 2 1 . • e 2 1 e 0 2 = ( ω 1 e 1 + ω 2 e 2 )( p 1 e 1 + p 2 e 2 )=( ω 1 p 1 + ω 2 p 2 c 2 ) e 2 1 = 0 . • Distinguish differe nt cases dep en ding on ω 1 p 1 + ω 2 p 2 c 2 is zero o r not. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Classification t w o-dimensional evolution algeb ras dim ( A 2 )=1 • When is B 0 a natural basis? • ( e 2 1 ) 2 = ω 2 1 e 2 1 + ω 2 2 e 2 2 = ω 2 1 e 2 1 + ω 2 2 ( c 2 e 2 1 )=( ω 2 1 + ω 2 2 c 2 ) e 2 1 . • e 0 2 2 = p 2 1 e 2 1 + p 2 2 e 2 2 = ( p 2 1 + p 2 2 c 2 ) e 2 1 . • e 2 1 e 0 2 = ( ω 1 e 1 + ω 2 e 2 )( p 1 e 1 + p 2 e 2 )=( ω 1 p 1 + ω 2 p 2 c 2 ) e 2 1 = 0 . • Distinguish differe nt cases dep en ding on ω 1 p 1 + ω 2 p 2 c 2 is zero o r not. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Classification t w o-dimensional evolution algeb ras Summa rizing T yp e A 2 has EP dim(ann(A)) A has a ideal b I  1 0 0 0  Y es 1 I = < e 1 >  1 − 1 1 − 1  No 0 No  1 1 0 0  Y es 0 I = < e 1 >  0 1 0 0  Y es 1 No EP = Extension p rop ert y b I is a non-degener a te p rincipal ideal. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Classification t w o-dimensional evolution algeb ras Summa rizing T yp e A 2 has EP dim(ann(A)) A has a ideal b I  1 0 0 0  Y es 1 I = < e 1 >  1 − 1 1 − 1  No 0 No  1 1 0 0  Y es 0 I = < e 1 >  0 1 0 0  Y es 1 No EP = Extension p rop ert y b I is a non-degener a te p rincipal ideal. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Classification t w o-dimensional evolution algeb ras Summa rizing T yp e A 2 has EP dim(ann(A)) A has a ideal b I  1 0 0 0  Y es 1 I = < e 1 >  1 − 1 1 − 1  No 0 No  1 1 0 0  Y es 0 I = < e 1 >  0 1 0 0  Y es 1 No EP = Extension p rop ert y b I is a non-degener a te p rincipal ideal. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Classification t w o-dimensional evolution algeb ras Summa rizing T yp e A 2 has EP dim(ann(A)) A has a ideal b I  1 0 0 0  Y es 1 I = < e 1 >  1 − 1 1 − 1  No 0 No  1 1 0 0  Y es 0 I = < e 1 >  0 1 0 0  Y es 1 No EP = Extension p rop ert y b I is a non-degener a te p rincipal ideal. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Classification t w o-dimensional evolution algeb ras Summa rizing T yp e A 2 has EP dim(ann(A)) A has a ideal b I  1 0 0 0  Y es 1 I = < e 1 >  1 − 1 1 − 1  No 0 No  1 1 0 0  Y es 0 I = < e 1 >  0 1 0 0  Y es 1 No EP = Extension p rop ert y b I is a non-degener a te p rincipal ideal. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Classification t w o-dimensional evolution algeb ras Summa rizing T yp e A 2 has EP dim(ann(A)) A has a ideal b I  1 0 0 0  Y es 1 I = < e 1 >  1 − 1 1 − 1  No 0 No  1 1 0 0  Y es 0 I = < e 1 >  0 1 0 0  Y es 1 No EP = Extension p rop ert y b I is a non-degener a te p rincipal ideal. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Classification t w o-dimensional evolution algeb ras Summa rizing T yp e A 2 has EP dim(ann(A)) A has a ideal b I  1 0 0 0  Y es 1 I = < e 1 >  1 − 1 1 − 1  No 0 No  1 1 0 0  Y es 0 I = < e 1 >  0 1 0 0  Y es 1 No EP = Extension p rop ert y b I is a non-degener a te p rincipal ideal. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Classification t w o-dimensional evolution algeb ras Summa rizing T yp e A 2 has EP dim(ann(A)) A has a ideal b I  1 0 0 0  Y es 1 I = < e 1 >  1 − 1 1 − 1  No 0 No  1 1 0 0  Y es 0 I = < e 1 >  0 1 0 0  Y es 1 No EP = Extension p rop ert y b I is a non-degener a te p rincipal ideal. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Classification t w o-dimensional evolution algeb ras Summa rizing T yp e A 2 has EP dim(ann(A)) A has a ideal b I  1 0 0 0  Y es 1 I = < e 1 >  1 − 1 1 − 1  No 0 No  1 1 0 0  Y es 0 I = < e 1 >  0 1 0 0  Y es 1 No EP = Extension p rop ert y b I is a non-degener a te p rincipal ideal. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Classification t w o-dimensional evolution algeb ras Summa rizing T yp e A 2 has EP dim(ann(A)) A has a ideal b I  1 0 0 0  Y es 1 I = < e 1 >  1 − 1 1 − 1  No 0 No  1 1 0 0  Y es 0 I = < e 1 >  0 1 0 0  Y es 1 No EP = Extension p rop ert y b I is a non-degenerate p rincipal ideal. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Classification t w o-dimensional evolution algeb ras dim ( A 2 )=2 • The numb er of non-zero entries in the structure matrix (main diagonal) is an inva riant. Then, w e have the follo wing p ossibilities: 1 0 0 1 ! , 1 α 0 1 ! , 1 α β 1 ! , 0 1 1 0 ! and 0 1 1 γ ! • W e s tu dy if the pa rametric families of evolution algeb ras a re isomo rphic when w e change of pa rameters. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Classification t w o-dimensional evolution algeb ras dim ( A 2 )=2 • The numb er of non-zero entries in the structure matrix (main diagonal) is an inva riant. Then, w e have the follo wing p ossibilities: 1 0 0 1 ! , 1 α 0 1 ! , 1 α β 1 ! , 0 1 1 0 ! and 0 1 1 γ ! • W e study if the pa rametric families of evolution algeb ras a re isomo rphic when w e change of pa rameters. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Classification of three-dimensional evolution algeb ras Outline 1 Intro duction 2 Basic facts ab out evolution algeb ras Evolution algeb ras Pro duct and Change of basis Subalgeb ras and ideals Non-degenerate evolution algeb ras The graph asso ciated to an evolution algeb ra 3 Decomp osition of an evolution algeb ra Ideals generated b y one element Simple evolution algeb ras Reducible evolution algeb ras The optimal direct-sum decomp osition of an evolution algeb ra 4 Classification t w o-dimensional evolution algeb ras 5 Classification of three-dimensional evolution algeb ras 6 F urther w o rk Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 A ction of S 3 o ( K × ) 3 on M 3 ( K ) 1 G =  α 0 0 0 β 0 0 0 γ  | α, β , γ ∈ K ×  =  ( α, β , γ ) | α, β , γ ∈ K ×  . 2 S 3 =  id 3 ,  0 1 0 1 0 0 0 0 1  ,  0 0 1 0 1 0 1 0 0  ,  1 0 0 0 0 1 0 1 0  ,  0 1 0 0 0 1 1 0 0  ,  0 0 1 1 0 0 0 1 0  3 H = { σ ( α , β , γ ) | σ ∈ S 3 , ( α , β , γ ) ∈ ( K × ) 3 } =  ( α, β , γ ) ,  0 α 0 β 0 0 0 0 γ  ,  0 0 α 0 β 0 γ 0 0  ,  α 0 0 0 0 β 0 γ 0  ,  0 α 0 0 0 β γ 0 0  ,  0 0 α β 0 0 0 γ 0  4 σ ( α 1 , α 2 , α 3 ) τ ( β 1 , β 2 , β 3 ) = σ τ ( α τ (1) β 1 , α τ (2) β 2 , α τ (3) β 3 ) . 5 Semidirect p ro duct of S 3 and ( K × ) 3 is denoted b y S 3 o ( K × ) 3 . 6 The action of P on M can b e fo rmulated as follo ws: P · M := P − 1 MP (2) . A ction of S 3 o ( K × ) 3 on M 3 ( K ) 1 G =  α 0 0 0 β 0 0 0 γ  | α, β , γ ∈ K ×  =  ( α, β , γ ) | α, β , γ ∈ K ×  . 2 S 3 =  id 3 ,  0 1 0 1 0 0 0 0 1  ,  0 0 1 0 1 0 1 0 0  ,  1 0 0 0 0 1 0 1 0  ,  0 1 0 0 0 1 1 0 0  ,  0 0 1 1 0 0 0 1 0  3 H = { σ ( α , β , γ ) | σ ∈ S 3 , ( α , β , γ ) ∈ ( K × ) 3 } =  ( α, β , γ ) ,  0 α 0 β 0 0 0 0 γ  ,  0 0 α 0 β 0 γ 0 0  ,  α 0 0 0 0 β 0 γ 0  ,  0 α 0 0 0 β γ 0 0  ,  0 0 α β 0 0 0 γ 0  4 σ ( α 1 , α 2 , α 3 ) τ ( β 1 , β 2 , β 3 ) = σ τ ( α τ (1) β 1 , α τ (2) β 2 , α τ (3) β 3 ) . 5 Semidirect p ro duct of S 3 and ( K × ) 3 is denoted b y S 3 o ( K × ) 3 . 6 The action of P on M can b e fo rmulated as follo ws: P · M := P − 1 MP (2) . A ction of S 3 o ( K × ) 3 on M 3 ( K ) 1 G =  α 0 0 0 β 0 0 0 γ  | α, β , γ ∈ K ×  =  ( α, β , γ ) | α, β , γ ∈ K ×  . 2 S 3 =  id 3 ,  0 1 0 1 0 0 0 0 1  ,  0 0 1 0 1 0 1 0 0  ,  1 0 0 0 0 1 0 1 0  ,  0 1 0 0 0 1 1 0 0  ,  0 0 1 1 0 0 0 1 0  3 H = { σ ( α , β , γ ) | σ ∈ S 3 , ( α , β , γ ) ∈ ( K × ) 3 } =  ( α, β , γ ) ,  0 α 0 β 0 0 0 0 γ  ,  0 0 α 0 β 0 γ 0 0  ,  α 0 0 0 0 β 0 γ 0  ,  0 α 0 0 0 β γ 0 0  ,  0 0 α β 0 0 0 γ 0  4 σ ( α 1 , α 2 , α 3 ) τ ( β 1 , β 2 , β 3 ) = σ τ ( α τ (1) β 1 , α τ (2) β 2 , α τ (3) β 3 ) . 5 Semidirect p ro duct of S 3 and ( K × ) 3 is denoted b y S 3 o ( K × ) 3 . 6 The action of P on M can b e fo rmulated as follo ws: P · M := P − 1 MP (2) . A ction of S 3 o ( K × ) 3 on M 3 ( K ) 1 G =  α 0 0 0 β 0 0 0 γ  | α, β , γ ∈ K ×  =  ( α, β , γ ) | α, β , γ ∈ K ×  . 2 S 3 =  id 3 ,  0 1 0 1 0 0 0 0 1  ,  0 0 1 0 1 0 1 0 0  ,  1 0 0 0 0 1 0 1 0  ,  0 1 0 0 0 1 1 0 0  ,  0 0 1 1 0 0 0 1 0  3 H = { σ ( α, β , γ ) | σ ∈ S 3 , ( α, β , γ ) ∈ ( K × ) 3 } =  ( α, β , γ ) ,  0 α 0 β 0 0 0 0 γ  ,  0 0 α 0 β 0 γ 0 0  ,  α 0 0 0 0 β 0 γ 0  ,  0 α 0 0 0 β γ 0 0  ,  0 0 α β 0 0 0 γ 0  4 σ ( α 1 , α 2 , α 3 ) τ ( β 1 , β 2 , β 3 ) = σ τ ( α τ (1) β 1 , α τ (2) β 2 , α τ (3) β 3 ) . 5 Semidirect p ro duct of S 3 and ( K × ) 3 is denoted b y S 3 o ( K × ) 3 . 6 The action of P on M can b e fo rmulated as follo ws: P · M := P − 1 MP (2) . A ction of S 3 o ( K × ) 3 on M 3 ( K ) 1 G =  α 0 0 0 β 0 0 0 γ  | α, β , γ ∈ K ×  =  ( α, β , γ ) | α, β , γ ∈ K ×  . 2 S 3 =  id 3 ,  0 1 0 1 0 0 0 0 1  ,  0 0 1 0 1 0 1 0 0  ,  1 0 0 0 0 1 0 1 0  ,  0 1 0 0 0 1 1 0 0  ,  0 0 1 1 0 0 0 1 0  3 H = { σ ( α, β , γ ) | σ ∈ S 3 , ( α, β , γ ) ∈ ( K × ) 3 } =  ( α, β , γ ) ,  0 α 0 β 0 0 0 0 γ  ,  0 0 α 0 β 0 γ 0 0  ,  α 0 0 0 0 β 0 γ 0  ,  0 α 0 0 0 β γ 0 0  ,  0 0 α β 0 0 0 γ 0  4 σ ( α 1 , α 2 , α 3 ) τ ( β 1 , β 2 , β 3 ) = σ τ ( α τ (1) β 1 , α τ (2) β 2 , α τ (3) β 3 ) . 5 Semidirect p ro duct of S 3 and ( K × ) 3 is denoted b y S 3 o ( K × ) 3 . 6 The action of P on M can b e fo rmulated as follo ws: P · M := P − 1 MP (2) . A ction of S 3 o ( K × ) 3 on M 3 ( K ) 1 G =  α 0 0 0 β 0 0 0 γ  | α, β , γ ∈ K ×  =  ( α, β , γ ) | α, β , γ ∈ K ×  . 2 S 3 =  id 3 ,  0 1 0 1 0 0 0 0 1  ,  0 0 1 0 1 0 1 0 0  ,  1 0 0 0 0 1 0 1 0  ,  0 1 0 0 0 1 1 0 0  ,  0 0 1 1 0 0 0 1 0  3 H = { σ ( α, β , γ ) | σ ∈ S 3 , ( α, β , γ ) ∈ ( K × ) 3 } =  ( α, β , γ ) ,  0 α 0 β 0 0 0 0 γ  ,  0 0 α 0 β 0 γ 0 0  ,  α 0 0 0 0 β 0 γ 0  ,  0 α 0 0 0 β γ 0 0  ,  0 0 α β 0 0 0 γ 0  4 σ ( α 1 , α 2 , α 3 ) τ ( β 1 , β 2 , β 3 ) = σ τ ( α τ (1) β 1 , α τ (2) β 2 , α τ (3) β 3 ) . 5 Semidirect p ro duct of S 3 and ( K × ) 3 is denoted b y S 3 o ( K × ) 3 . 6 The action of P on M can b e fo rmulated as follo ws: P · M := P − 1 MP (2) . A ction of S 3 o ( K × ) 3 on M 3 ( K ) 1 G =  α 0 0 0 β 0 0 0 γ  | α, β , γ ∈ K ×  =  ( α, β , γ ) | α, β , γ ∈ K ×  . 2 S 3 =  id 3 ,  0 1 0 1 0 0 0 0 1  ,  0 0 1 0 1 0 1 0 0  ,  1 0 0 0 0 1 0 1 0  ,  0 1 0 0 0 1 1 0 0  ,  0 0 1 1 0 0 0 1 0  3 H = { σ ( α, β , γ ) | σ ∈ S 3 , ( α, β , γ ) ∈ ( K × ) 3 } =  ( α, β , γ ) ,  0 α 0 β 0 0 0 0 γ  ,  0 0 α 0 β 0 γ 0 0  ,  α 0 0 0 0 β 0 γ 0  ,  0 α 0 0 0 β γ 0 0  ,  0 0 α β 0 0 0 γ 0  4 σ ( α 1 , α 2 , α 3 ) τ ( β 1 , β 2 , β 3 ) = σ τ ( α τ (1) β 1 , α τ (2) β 2 , α τ (3) β 3 ) . 5 Semidirect p ro duct of S 3 and ( K × ) 3 is denoted b y S 3 o ( K × ) 3 . 6 The action of P on M can b e fo rmulated as follo ws: P · M := P − 1 MP (2) . Evolution Algeb ras Classification of three-dimensional evolution algeb ras Prop osition F o r any P ∈ S 3 o ( K × ) 3 and any M ∈ M 3 ( K ) w e have: 1 The numb er of zero entries in M coincides with the numb er of zero entries in P · M . 2 The numb er of zero entries in the main diagonal of M coincides with the numb er of zero entries in the main diagonal of P · M . 3 The rank of M and the rank of P · M coincide. 4 Let M B b e the structure matrix of an evolution algeb ra A such that A 2 = A . If N is the structure matrix of A relative to a natural basis B 0 then there exists Q ∈ S 3 o ( K × ) 3 such that N = Q · M B . Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Classification of three-dimensional evolution algeb ras Three-dimensional evolution algeb ras Let A b e a three-dimensional evolution K -algeb ra whe re K is a field of cha racteristic different from 2 and such that fo r any k ∈ K the p olynomial of the fo rm x n − k has a ro ot whenever n = 2 , 3 , 7. X If dim ( A 2 ) = 0 then M B = 0 fo r any natural basis B of A . X If dim ( A 2 ) = 1. Let M B = ( ω ij ) b e the structu re matrix. • W e m a y assume e 2 1 6 = 0. e 2 1 = ω 1 e 1 + ω 2 e 2 + ω 3 e 3 e 2 2 = c 1 e 2 1 = c 1 ( ω 1 e 1 + ω 2 e 2 + ω 3 e 3 ) e 2 3 = c 2 e 2 1 = c 2 ( ω 1 e 1 + ω 2 e 2 + ω 3 e 3 ) . Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Classification of three-dimensional evolution algeb ras Three-dimensional evolution algeb ras Let A b e a three-dimensional evolution K -algeb ra where K is a field of cha racteristic different from 2 and such that fo r any k ∈ K the p olynomial of the fo rm x n − k has a ro ot whenever n = 2 , 3 , 7. X If dim ( A 2 ) = 0 then M B = 0 fo r any natural basis B of A . X If dim ( A 2 ) = 1. Let M B = ( ω ij ) b e the structu re matrix. • W e m a y assume e 2 1 6 = 0. e 2 1 = ω 1 e 1 + ω 2 e 2 + ω 3 e 3 e 2 2 = c 1 e 2 1 = c 1 ( ω 1 e 1 + ω 2 e 2 + ω 3 e 3 ) e 2 3 = c 2 e 2 1 = c 2 ( ω 1 e 1 + ω 2 e 2 + ω 3 e 3 ) . Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Classification of three-dimensional evolution algeb ras Three-dimensional evolution algeb ras Let A b e a three-dimensional evolution K -algeb ra where K is a field of cha racteristic different from 2 and such that for any k ∈ K the p olynomial of the fo rm x n − k has a ro ot whenever n = 2 , 3 , 7. X If dim ( A 2 ) = 0 then M B = 0 fo r any natural basis B of A . X If dim ( A 2 ) = 1. Let M B = ( ω ij ) b e the structu re matrix. • W e m a y assume e 2 1 6 = 0. e 2 1 = ω 1 e 1 + ω 2 e 2 + ω 3 e 3 e 2 2 = c 1 e 2 1 = c 1 ( ω 1 e 1 + ω 2 e 2 + ω 3 e 3 ) e 2 3 = c 2 e 2 1 = c 2 ( ω 1 e 1 + ω 2 e 2 + ω 3 e 3 ) . Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Classification of three-dimensional evolution algeb ras Three-dimensional evolution algeb ras Let A b e a three-dimensional evolution K -algeb ra where K is a field of cha racteristic different from 2 and such that for any k ∈ K the p olynomial of the fo rm x n − k has a ro ot whenever n = 2 , 3 , 7. X If dim ( A 2 ) = 0 then M B = 0 for any natural basis B of A . X If dim ( A 2 ) = 1. Let M B = ( ω ij ) b e the structu re matrix. • W e m a y assume e 2 1 6 = 0. e 2 1 = ω 1 e 1 + ω 2 e 2 + ω 3 e 3 e 2 2 = c 1 e 2 1 = c 1 ( ω 1 e 1 + ω 2 e 2 + ω 3 e 3 ) e 2 3 = c 2 e 2 1 = c 2 ( ω 1 e 1 + ω 2 e 2 + ω 3 e 3 ) . Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Classification of three-dimensional evolution algeb ras Three-dimensional evolution algeb ras Let A b e a three-dimensional evolution K -algeb ra where K is a field of cha racteristic different from 2 and such that for any k ∈ K the p olynomial of the fo rm x n − k has a ro ot whenever n = 2 , 3 , 7. X If dim ( A 2 ) = 0 then M B = 0 fo r any natural basis B of A . X If dim ( A 2 ) = 1. Let M B = ( ω ij ) b e the structure matrix. • W e m a y assume e 2 1 6 = 0. e 2 1 = ω 1 e 1 + ω 2 e 2 + ω 3 e 3 e 2 2 = c 1 e 2 1 = c 1 ( ω 1 e 1 + ω 2 e 2 + ω 3 e 3 ) e 2 3 = c 2 e 2 1 = c 2 ( ω 1 e 1 + ω 2 e 2 + ω 3 e 3 ) . Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Classification of three-dimensional evolution algeb ras Three-dimensional evolution algeb ras Let A b e a three-dimensional evolution K -algeb ra where K is a field of cha racteristic different from 2 and such that for any k ∈ K the p olynomial of the fo rm x n − k has a ro ot whenever n = 2 , 3 , 7. X If dim ( A 2 ) = 0 then M B = 0 fo r any natural basis B of A . X If dim ( A 2 ) = 1. Let M B = ( ω ij ) b e the structure matrix. • W e ma y assume e 2 1 6 = 0. e 2 1 = ω 1 e 1 + ω 2 e 2 + ω 3 e 3 e 2 2 = c 1 e 2 1 = c 1 ( ω 1 e 1 + ω 2 e 2 + ω 3 e 3 ) e 2 3 = c 2 e 2 1 = c 2 ( ω 1 e 1 + ω 2 e 2 + ω 3 e 3 ) . Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Classification of three-dimensional evolution algeb ras • W e a n a lyze when A 2 has the extension p rop ert y , i.e., if there exists a natural basis B 0 = { e 2 1 , e 0 2 , e 0 3 } of A with P B 0 B =    ω 1 α δ ω 2 β ν ω 3 γ η    • The conditions a re as follo ws: α ω 1 + β ω 2 c 1 + γ ω 3 c 2 = 0 δ ω 1 + ν ω 2 c 1 + η ω 3 c 2 = 0 α δ + β ν c 1 + γ η c 2 = 0 | P B 0 B | 6 = 0 • A 2 has the extension p rop ert y if and only if ω 2 1 + ω 2 2 c 1 + ω 2 3 c 2 6 = 0 . Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Classification of three-dimensional evolution algeb ras • W e analyze when A 2 has the extension p rop ert y , i.e., if there exists a natural basis B 0 = { e 2 1 , e 0 2 , e 0 3 } of A with P B 0 B =    ω 1 α δ ω 2 β ν ω 3 γ η    • The conditions a re as follo ws: α ω 1 + β ω 2 c 1 + γ ω 3 c 2 = 0 δ ω 1 + ν ω 2 c 1 + η ω 3 c 2 = 0 α δ + β ν c 1 + γ η c 2 = 0 | P B 0 B | 6 = 0 • A 2 has the extension p rop ert y if and only if ω 2 1 + ω 2 2 c 1 + ω 2 3 c 2 6 = 0 . Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Classification of three-dimensional evolution algeb ras • W e analyze when A 2 has the extension p rop ert y , i.e., if there exists a natural basis B 0 = { e 2 1 , e 0 2 , e 0 3 } of A with P B 0 B =    ω 1 α δ ω 2 β ν ω 3 γ η    • The conditions a re as follo ws: αω 1 + β ω 2 c 1 + γ ω 3 c 2 = 0 δ ω 1 + ν ω 2 c 1 + η ω 3 c 2 = 0 αδ + β ν c 1 + γ η c 2 = 0 | P B 0 B | 6 = 0 • A 2 has the extension p rop ert y if and only if ω 2 1 + ω 2 2 c 1 + ω 2 3 c 2 6 = 0 . Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 dim ( A 2 )=1 T ype A 2 has the extension dimension of A has a principal degenerate prop ert y ann(A) t w o-dimensional ev ol ution ideal  1 − 1 1 1 − 1 1 000  No 0 I = < e 3 >  1 − 1 0 1 − 1 0 000  No 1 I = < e 1 + e 2 + e 3 >  1 1 1 0 0 0 0 0 0  Y es 0 No  1 0 1 0 0 0 0 0 0  Y e s 1 No  1 0 0 0 0 0 0 0 0  Y e s 2 No  0 1 1 0 0 0 0 0 0  Y e s 1 I = < e 3 >  0 0 1 0 0 0 0 0 0  Y e s 2 I = < e 3 > dim ( A 2 )=1 T ype A 2 has the extension dimension of A has a principal degenerate prop ert y ann(A) t w o-dimensional ev ol ution ideal  1 − 1 1 1 − 1 1 000  No 0 I = < e 3 >  1 − 1 0 1 − 1 0 000  No 1 I = < e 1 + e 2 + e 3 >  1 1 1 0 0 0 0 0 0  Y es 0 No  1 0 1 0 0 0 0 0 0  Y es 1 No  1 0 0 0 0 0 0 0 0  Y e s 2 No  0 1 1 0 0 0 0 0 0  Y e s 1 I = < e 3 >  0 0 1 0 0 0 0 0 0  Y e s 2 I = < e 3 > dim ( A 2 )=1 T ype A 2 has the extension dimension of A has a principal degenerate prop ert y ann(A) t w o-dimensional ev ol ution ideal  1 − 1 1 1 − 1 1 000  No 0 I = < e 3 >  1 − 1 0 1 − 1 0 000  No 1 I = < e 1 + e 2 + e 3 >  1 1 1 0 0 0 0 0 0  Y es 0 No  1 0 1 0 0 0 0 0 0  Y es 1 No  1 0 0 0 0 0 0 0 0  Y es 2 No  0 1 1 0 0 0 0 0 0  Y e s 1 I = < e 3 >  0 0 1 0 0 0 0 0 0  Y e s 2 I = < e 3 > dim ( A 2 )=1 T ype A 2 has the extension dimension of A has a principal degenerate prop ert y ann(A) t w o-dimensional ev ol ution ideal  1 − 1 1 1 − 1 1 000  No 0 I = < e 3 >  1 − 1 0 1 − 1 0 000  No 1 I = < e 1 + e 2 + e 3 >  1 1 1 0 0 0 0 0 0  Y es 0 No  1 0 1 0 0 0 0 0 0  Y es 1 No  1 0 0 0 0 0 0 0 0  Y es 2 No  0 1 1 0 0 0 0 0 0  Y es 1 I = < e 3 >  0 0 1 0 0 0 0 0 0  Y e s 2 I = < e 3 > dim ( A 2 )=1 T ype A 2 has the extension dimension of A has a principal degenerate prop ert y ann(A) t w o-dimensional ev ol ution ideal  1 − 1 1 1 − 1 1 000  No 0 I = < e 3 >  1 − 1 0 1 − 1 0 000  No 1 I = < e 1 + e 2 + e 3 >  1 1 1 0 0 0 0 0 0  Y es 0 No  1 0 1 0 0 0 0 0 0  Y es 1 No  1 0 0 0 0 0 0 0 0  Y es 2 No  0 1 1 0 0 0 0 0 0  Y es 1 I = < e 3 >  0 0 1 0 0 0 0 0 0  Y es 2 I = < e 3 > dim ( A 2 )=1 T ype A 2 has the extension dimension of A has a principal degenerate prop ert y ann(A) t w o-dimensional ev ol ution ideal  1 − 1 1 1 − 1 1 000  No 0 I = < e 3 >  1 − 1 0 1 − 1 0 000  No 1 I = < e 1 + e 2 + e 3 >  1 1 1 0 0 0 0 0 0  Y es 0 No  1 0 1 0 0 0 0 0 0  Y es 1 No  1 0 0 0 0 0 0 0 0  Y es 2 No  0 1 1 0 0 0 0 0 0  Y es 1 I = < e 3 >  0 0 1 0 0 0 0 0 0  Y es 2 I = < e 3 > dim ( A 2 )=1 T ype A 2 has the extension dimension of A has a principal degenerate prop ert y ann(A) t w o-dimensional ev ol ution ideal  1 − 1 1 1 − 1 1 000  No 0 I = < e 3 >  1 − 1 0 1 − 1 0 000  No 1 I = < e 1 + e 2 + e 3 >  1 1 1 0 0 0 0 0 0  Y es 0 No  1 0 1 0 0 0 0 0 0  Y es 1 No  1 0 0 0 0 0 0 0 0  Y es 2 No  0 1 1 0 0 0 0 0 0  Y es 1 I = < e 3 >  0 0 1 0 0 0 0 0 0  Y es 2 I = < e 3 > dim ( A 2 )=1 T ype A 2 has the extension dimension of A has a principal degenerate prop ert y ann(A) t w o-dimensional ev ol ution ideal  1 − 1 1 1 − 1 1 000  No 0 I = < e 3 >  1 − 1 0 1 − 1 0 000  No 1 I = < e 1 + e 2 + e 3 >  1 1 1 0 0 0 0 0 0  Y es 0 No  1 0 1 0 0 0 0 0 0  Y es 1 No  1 0 0 0 0 0 0 0 0  Y es 2 No  0 1 1 0 0 0 0 0 0  Y es 1 I = < e 3 >  0 0 1 0 0 0 0 0 0  Y es 2 I = < e 3 > dim ( A 2 )=1 T ype A 2 has the extension dimension of A has a principal degenerate prop ert y ann(A) t w o-dimensional ev ol ution ideal  1 − 1 1 1 − 1 1 000  No 0 I = < e 3 >  1 − 1 0 1 − 1 0 000  No 1 I = < e 1 + e 2 + e 3 >  1 1 1 0 0 0 0 0 0  Y es 0 No  1 0 1 0 0 0 0 0 0  Y es 1 No  1 0 0 0 0 0 0 0 0  Y es 2 No  0 1 1 0 0 0 0 0 0  Y es 1 I = < e 3 >  0 0 1 0 0 0 0 0 0  Y es 2 I = < e 3 > dim ( A 2 )=1 T ype A 2 has the extension dimension of A has a principal degenerate prop ert y ann(A) t w o-dimensional ev ol ution ideal  1 − 1 1 1 − 1 1 000  No 0 I = < e 3 >  1 − 1 0 1 − 1 0 000  No 1 I = < e 1 + e 2 + e 3 >  1 1 1 0 0 0 0 0 0  Y es 0 No  1 0 1 0 0 0 0 0 0  Y es 1 No  1 0 0 0 0 0 0 0 0  Y es 2 No  0 1 1 0 0 0 0 0 0  Y es 1 I = < e 3 >  0 0 1 0 0 0 0 0 0  Y es 2 I = < e 3 > Evolution Algeb ras Classification of three-dimensional evolution algeb ras c 1 = c 2 = 0 ♣ If c 1 = c 2 = 0. • The p ossible change of basis matrices a re:  p 11 0 0 0 p 22 0 p 31 p 32 p 33  ,  0 p 12 0 p 21 0 0 p 31 p 32 p 33  . • The numb er of non-zero entries in the first and second ro ws is p reserved. • The only p ossibilities a re:  ω 11 0 0 0 ω 22 0 0 0 0  ,  0 ω 12 0 ω 21 0 0 0 0 0  ,  ω 11 0 0 ω 21 ω 22 0 0 0 0  ,  0 ω 12 0 ω 21 ω 22 0 0 0 0  ,  ω 11 ω 12 0 ω 21 ω 22 0 0 0 0  S  ω 11 0 0 0 0 0 0 ω 32 0  ,  ω 11 ω 12 0 0 0 0 0 ω 32 0  ,  ω 11 0 0 ω 21 0 0 0 ω 32 0   0 ω 12 0 0 0 0 ω 31 0 0  • W e mak e as many ones as p ossible . • Are the pa rametric families evolution algeb ras isomo r phic?. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Classification of three-dimensional evolution algeb ras c 1 = c 2 = 0 ♣ If c 1 = c 2 = 0. • The p ossible change of basis matrices a re:  p 11 0 0 0 p 22 0 p 31 p 32 p 33  ,  0 p 12 0 p 21 0 0 p 31 p 32 p 33  . • The numb er of non-zero entries in the first and second ro ws is p reserved. • The only p ossibilities a re:  ω 11 0 0 0 ω 22 0 0 0 0  ,  0 ω 12 0 ω 21 0 0 0 0 0  ,  ω 11 0 0 ω 21 ω 22 0 0 0 0  ,  0 ω 12 0 ω 21 ω 22 0 0 0 0  ,  ω 11 ω 12 0 ω 21 ω 22 0 0 0 0  S  ω 11 0 0 0 0 0 0 ω 32 0  ,  ω 11 ω 12 0 0 0 0 0 ω 32 0  ,  ω 11 0 0 ω 21 0 0 0 ω 32 0   0 ω 12 0 0 0 0 ω 31 0 0  • W e mak e as many ones as p ossible. • Are the pa rametric families evolution algeb ras isomo rphic? . Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Classification of three-dimensional evolution algeb ras dim ( A 2 )=3 X If dim ( A 2 ) = 3. • The only ch a nge of basis matrices a re S 3 o ( K × ) 3 . • The numb er of non-zero entries is inva riant. • The minimum numb er of non-zero entries is three. • W e m ak e as m any ones as p ossible. • W e s tu dy when the pa ram e tric families of evolution algeb ras a re is omo rphic. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Classification of three-dimensional evolution algeb ras dim ( A 2 )=3 X If dim ( A 2 ) = 3. • The only ch a nge of basis matrices a re S 3 o ( K × ) 3 . • The numb er of non-zero entries is inva riant. • The minimum numb er of non-zero entries is three. • W e m ak e as m any ones as p ossible. • W e s tu dy when the pa ram e tric families of evolution algeb ras a re is omo rphic. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Classification of three-dimensional evolution algeb ras dim ( A 2 )=3 X If dim ( A 2 ) = 3. • The only change of basis matrices a re S 3 o ( K × ) 3 . • The numb er of non-zero entries is inva riant. • The minimum numb er of non-zero entries is three. • W e m ak e as m any ones as p ossible. • W e s tu dy when the pa ram e tric families of evolution algeb ras a re is omo rphic. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Classification of three-dimensional evolution algeb ras dim ( A 2 )=3 X If dim ( A 2 ) = 3. • The only change of basis matrices a re S 3 o ( K × ) 3 . • The numb er of non-zero entries is inva riant . • The minimum numb er of non-zero entries is three. • W e m ak e as m any ones as p ossible. • W e s tu dy when the pa ram e tric families of evolution algeb ras a re is omo rphic. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Classification of three-dimensional evolution algeb ras dim ( A 2 )=3 X If dim ( A 2 ) = 3. • The only change of basis matrices a re S 3 o ( K × ) 3 . • The numb er of non-zero entries is inva riant. • The minimum numb er of non-zero entries is three . • W e m ak e as m any ones as p ossible. • W e s tu dy when the pa ram e tric families of evolution algeb ras a re is omo rphic. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Classification of three-dimensional evolution algeb ras dim ( A 2 )=3 X If dim ( A 2 ) = 3. • The only change of basis matrices a re S 3 o ( K × ) 3 . • The numb er of non-zero entries is inva riant. • The minimum numb er of non-zero entries is three. • W e mak e as many ones as p ossible . • W e s tu dy when the pa ram e tric families of evolution algeb ras a re is omo rphic. Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras Classification of three-dimensional evolution algeb ras dim ( A 2 )=3 X If dim ( A 2 ) = 3. • The only change of basis matrices a re S 3 o ( K × ) 3 . • The numb er of non-zero entries is inva riant. • The minimum numb er of non-zero entries is three. • W e mak e as many ones as p ossible. • W e study when the pa rametric families of evolution algeb ras a re isomo rphic . Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras F urther w o rk F urther w o rks • Another p rop erties from the evolution algeb ras that can b e read in terms of its underlying graph, and conversely. • Will the optimal direct-sum decomp osition have an impact from the biological p oint of view?. • Biological application of the classification of evolution algeb ras • Classification of the alternative evolution algeb ras. • The different metho ds use to obtain the classification can b e generalized to a rbitra ry finite-dimensional evolution algeb ras . Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017 Evolution Algeb ras F urther w o rk “ The lack of real contact b et w een mathematics and biology is either a tragedy , a scandal o r a challenge, it is ha rd to decide which. ” Gian Ca rlo Rota Thanks! Y olanda Cab rera Casado (Universidad de M´ alaga) 16 Decemb er 2017