SOME PROPERTIES OF EVOLUTION ALGEBRAS
L. M. CAMACHO, J. R. G´
OMEZ, B. A. OMIROV, R. M. TURDIBAEV
Abs ac . The pape is de o ed o he s udy o ini e dimensional complex e olu-
ion algeb as. The class o e olu ion algeb as isomo phic o e olu ion algeb as wi h
Jo dan o m ma ices is desc ibed. Fo ini e dimensional complex e olu ion algeb as
he c i e ia o nilpo ency is es ablished in e ms o he p ope ies o co esponding
ma ices. Mo eo e , i is p o ed ha o nilpo en n−dimensional complex e olu ion
algeb as he possible maximal nilpo ency index is 1 + 2n−1.The c i e ia o plana i y
o ini e g aphs is o mula ed by means o e olu ion algeb as de ined by g aphs.
AMS classi ica ions (2010): 05C25; 17A36; 17D92; 17D99
Keywo ds: E olu ion algeb a; commu a i e algeb a; isomo phism; nilpo ency; pla-
na g aph.
1. In oduc ion
In 20s and 30s o he las cen u y he new objec was in oduced o ma hema ics,
which was he p oduc o in e ac ions be ween Mendelian gene ics and ma hema ics.
Mendel es ablished he basic laws o inhe i ance, which a e summa ized as Mendel’s
Law o Seg ega ion and Mendel’s Law o Independen Asso men . This laws we e
ma hema ically o mula ed by Se eb owsky [9], who was also he i s o gi e an al-
geb aic in e p e a ion o he ” ×” sign, which indica ed sexual ep oduc ion. La e
Gli enko [5] used he no ion o Mendelian algeb as in his wo k. Also Kos i zin [7]
independen ly in oduced a ”symbolic mul iplica ion” o exp ess Mendel’s laws. In his
se e al pape s E he ing on [2]- [4] in oduced he o mal language o abs ac algeb a
o he s udy o he gene ics. These algeb as, in gene al, a e non-associa i e.
Howe e , in he beginning o he XX cen u y in gene ics he e we e disco e ed se -
e al examples o inhe i ances, whe e ai s do no seg ega e in acco dance wi h Mendel’s
laws. In he p esen day, non-Mendelian gene ics is a basic language o molecula ge-
ne ics. Non-Mendelian inhe i ance plays an impo an ole in se e al disease p ocesses.
Na u ally, he ques ion a ises: wha non-Mendelian gene ics o e s o ma hema ics?
The e olu ion algeb as, in oduced in [10] se es as he answe o his ques ion.
The concep o e olu ion algeb as lies be ween algeb as and dynamical sys ems. Al-
geb aically, e olu ion algeb as a e non-associa i e Banach algeb a; dynamically, hey
ep esen disc e e dynamical sys ems. E olu ion algeb as ha e many connec ions wi h
a ious b anches o ma hema ics, such as g aph heo y, g oup heo y, s ochas ic p o-
cesses, ma hema ical physics e c. Since e olu ion algeb as a e no de ined by iden i ies,
hey can no belong o any well-known classes o non-associa i e algeb as, as Lie, al-
e na i e and Jo dan algeb as.
The ounda ion o e olu ion algeb a heo y and applica ions in non-Mendelian ge-
ne ics and Ma ko chains a e de eloped, wi h poin e s o some u he esea ch opics
was gi en in book [11].
In his pape , we s udy some p ope ies o ini e dimensional complex e olu ion alge-
b as. Since any e olu ion algeb a in a na u al basis is de ined by a quad a ic ma ix, we
1
2 L. M. CAMACHO, J. R. G ´
OMEZ, B. A. OMIROV, R. M. TURDIBAEV
s udy he connec ion be ween he algeb aic s uc u e o e olu ion algeb as and ma i-
ces. Mo e p ecise esul s a e ob ained o e olu ion algeb as wi h non-singula ma ices.
Fo example, he only au omo phisms o such algeb as a e he composi ion o basis
pe mu a ion and he mul iplica ion o basic ec o s o scala s. Since in he ma ix
heo y he Jo dan o m o he ma ix is essen ial opic, we in es iga e a class o e o-
lu ion algeb as isomo phic o e olu ion algeb as wi h Jo dan o m ma ices. Thus we
can dis inguish he class o e olu ion algeb as wi h a ma ix in which he eigen alues
a e known. The e o e, co esponding algeb as can be in es iga ed by he eigen alues
in algeb aical poin o iew. Namely, he p oblem o econs uc ion o Ma ko chains
on ees [8] which depends on he second eigen alue can be s udied by abo e e olu ion
algeb as.
In [4] i was poin ed ou o gene al gene ic algeb as ha he nilpo en p ope y is
essen ial o hese algeb as and he de ini ion as ain algeb as and ba ic algeb as we e
o mula ed. By his means, we de ine nil, sol able, igh -nilpo en and nilpo en e olu-
ion algeb as as in [1] and s udy some p ope ies o n−dimensional nilpo en e olu ion
algeb as. The no ions as igh nilpo ency and nili y o ini e dimensional e olu ion
algeb as a e equi alen [1]. In his wo k, we p o e ha any n−dimensional igh -
nilpo en e olu ion algeb a is nilpo en . Mo eo e , o e olu ion algeb as o dimension
nwe desc ibe some possible alues o indexes o nilpo ency and p o e ha 1 + 2n−1is
a maximal nilpo ency index.
In [11] he ela ion be ween g aph heo y and e olu ion algeb as was gi en. The
las sec ion o his wo k is dedica ed o he s udy o some e olu ion algeb as de ined
by g aphs, namely we ind some algeb aic p ope ies o e olu ion algeb as de ined by
comple e and comple e bipa i e g aphs and e o mula e he g aph plana i y c i e ia in
e ms o e olu ion algeb as.
2. P elimina ies
Now we de ine he main objec o he pape .
De ini ion 2.1. [11] Le Ebe a ec o space o e a ield Kwi h de ined mul iplica ion
·and a basis {e1, e2,...}such ha
ei·ej= 0, i 6=j,
ei·ei=X
k
aikek, i ≥1,
hen Eis called e olu ion algeb a and basis {e1, e2,...}is said o be na u al basis.
F om he abo e de ini ion i ollows ha e olu ion algeb as a e commu a i e ( he e-
o e, lexible).
Le Ebe a ini e dimensional e olu ion algeb a wi h na u al basis {e1,...,en}, hen
ei·ei=
n
X
j=1
aijej,1≤i≤n,
whe e emaining p oduc s a e equal o ze o.
The ma ix A= (aij )n
i,j=1 is called ma ix o he algeb a Ein na u al basis
{e1,...,en}.
In [11] condi ions o basis ans o ma ions ha p ese e na u alness o he basis
a e gi en. Also, he ela ion be ween he ma ices in a new and old na u al basis is
es ablished in e ms o new de ined ope a ion on ma ices. Since his app oach is no
SOME PROPERTIES OF EVOLUTION ALGEBRAS 3
p ac ical o ou u he pu poses, below we gi e he ollowing b ie e sion in e ms o
i s ma ix elemen s.
Now le us conside non-singula linea ans o ma ion o he gi en na u al basis
{e1,...,en}by ma ix T= ( ij)n
i,j=1 :
i=
n
X
j=1
ijej,1≤i≤n.
This ans o ma ion is isomo phism i and only i i j= 0 o all i6=j.
Thus,
i· j=
n
X
p=1
ip jp(ep·ep) =
n
X
k=1 n
X
p=1
ip jpapk!ek= 0.
Hence, i Tis an isomo phism, hen o i6=jand 1 ≤k≤nwe ha e
n
X
p=1
ip jpapk = 0.(2.1)
Obse e ha
i· i=
n
X
p=1
2
ip(ep·ep) =
n
X
p=1
2
ip
n
X
k=1
apkek=
n
X
k=1 n
X
p=1
2
ipapk!ek.
Now le Tij be he elemen s o ma ix T−1.Then ek=
n
X
s=1
Tks sand
i· i=
n
X
k=1 n
X
p=1
2
ipapk!n
X
s=1
Tks s=
n
X
s=1 n
X
k=1
n
X
p=1
2
ipapkTks! s.
Hence, o he elemen s o he ma ix B= (bis)i,s=1,n o e olu ion algeb a Ein na u al
basis { 1,..., n}we ha e
bis =
n
X
k=1
n
X
p=1
2
ipapkTks.(2.2)
De ini ion 2.2. An elemen ao e olu ion algeb a Eis called nil i he e exis s n(a)∈N
such ha (...((a·a)·a)·...)·a)
|{z }
n(a) imes
= 0.E olu ion algeb a Eis called nil i any elemen
o he algeb a is nil.
We in oduce he ollowing sequences:
E(1) =E, E(k+1) =E(k)E(k), k ≥1
E<1>=E, E<k+1>=E<k>E, k ≥1
E1=E, Ek=
k−1
X
i=1
EiEk−i, k ≥1
No e ha is no di icul o p o e he ollowing inclusions o k≥1 :
E<k> ⊆Ek, E(k+1) ⊆E2k.
4 L. M. CAMACHO, J. R. G ´
OMEZ, B. A. OMIROV, R. M. TURDIBAEV
Also, no e ha since Eis commu a i e algeb a we ob ain Ek=X
1≤i≤k−i
EiEk−i.
De ini ion 2.3. An e olu ion algeb a Eis called
(i) sol able i he e exis s n∈Nsuch ha E(n)= 0 and he minimal such numbe is
called index o sol abili y;
(ii) igh nilpo en i he e exis s n∈Nsuch ha E<n> = 0 and he minimal such
numbe is called index o igh nilpo ency;
(iii) nilpo en i he e exis s n∈Nsuch ha En= 0 and he minimal such numbe
is called index o nilpo ency.
Obse e ha i e olu ion algeb a is nilpo en , hen i is igh nilpo en and sol able.
The ollowing example shows ha sol able e olu ion algeb a is no necessa ily a igh
nilpo en algeb a.
Example 2.4. Le Ebe an e olu ion algeb a wi h na u al basis {e1,...en}and he
ollowing mul iplica ion:
eiei=e1+···+en,1≤i≤n−1
enen= (1 −n)(e1+···+en).
Then E(3) = 0,bu Ek=he1+···+eni o k≥2.
The example desc ibed abo e in ac is a pa icula case o he ollowing
P oposi ion 2.5. p o-sol Le Ebe an n−dimensional complex e olu ion algeb a such
ha dim E(2) = 1.Then E(3) = 0 i and only i Eis isomo phic o an e olu ion algeb a
wi h na u al basis {e1,...,en}wi h he ollowing mul iplica ion:
eiei=λi(e1+···+ek),1≤i≤n,
whe e λi∈C,
k
X
j=1
λj= 0,
n
X
j=1
|λj|26= 0 and 1≤k≤n.
P oo . Since E(2) = 1 and E(2) is spanned by eiei,1≤i≤nwe ob ain ha hey a e
collinea o a non-ze o ec o x=a1e1+···+anen. Wi h he sui able na u al basis
change, one can assume ha x=e1+···+ek o some 1 ≤k≤n.
Le eiei=λix, 1≤i≤nand
n
X
j=1
|λj|26= 0.Then E(3) is spanned by
xx = (e1+···+ek)2=
k
X
j=1
λjx.
Hence, E(3) = 0 i and only i
k
X
j=1
λj= 0.
Rema k 2.6. Ac ually, he mul iplica ion ob ained in P oposi ion ?? can be di ided
in o wo disjoin classes. Fi s one, when λi= 0 o all 1≤i≤k, hen his e olu ion
algeb a is nilpo en . The second one is when λi6= 0 o some 1≤i≤k. Then by
na u al basis ans o ma ion one can assume ha e1e1=e1+···+ekand hence, his
e olu ion algeb a is no nilpo en .
In [1] he equi alence o igh nilpo ency and nili y o ini e dimensional complex
e olu ion algeb as is p o ed.
SOME PROPERTIES OF EVOLUTION ALGEBRAS 5
Theo em 2.7. The ollowing s a emen s a e equi alen :
a) The ma ix o an e olu ion algeb a Ecan be ans o med by na u al basis pe mu-
a ion o
A=





0a12 a13 . . . a1n
0 0 a23 . . . a2n
0 0 0 . . . a3n
.
.
..
.
..
.
.....
.
.
0 0 0 ... 0





; (2.3)
b) E olu ion algeb a Eis igh nilpo en algeb a;
c) E olu ion algeb a Eis nil algeb a.
3. Isomo phisms
In case o e olu ion algeb as wi h non-singula e olu ion ma ices he p oblem o
inding isomo phic algeb as o he gi en one can be sol ed mo e p ecisely.
Le Ebe an e olu ion algeb a wi h ma ix Asuch ha de A6= 0.
P oposi ion 3.1. Au (E) = {Tπ|π∈Sn},whe e Tπ= ( ij )1≤i,j≤nsuch ha ij 6= 0 i
and only i j=π(i).Mo eo e , i Tπis an au omo phism o e olu ion algeb a Eand
B= (bij)1≤i,j≤nis he ma ix o Ein basis Tπ(e1),...,Tπ(en) hen
bij = 2
i,π(i)
j,π(j)
·aπ(i)π(j).(3.1)
P oo . Conside (2.1) as a linea homogeneous sys em o equa ions in e ms o unknowns
i1 j1,..., in jn.I Ais a non-singula ma ix hen om (2.1) we ob ain







i1 j1= 0
i2 j2= 0
...
in jn = 0
whe e i6=j.
Since ma ix Tis non-singula , in e e y ow he e is a leas one non-ze o elemen .
Bu o any non-ze o elemen ip (in he i− h ow) we ha e ip jp = 0 o all j6=i.
The e o e, jp = 0 o j6=i. Now i o some m6=pwe ha e im 6= 0, hen simila ly, we
ob ain jm o all j6=m. Bu his con adic s o non-singula i y o ma ix T. The e o e,
in e e y ow and e e y column we ha e exac ly one non-ze o elemen , i.e., he ma ix
Thas he o m desc ibed in he s a emen o he p oposi ion.
No e ha de T= (−1)σ(π) 1π(1) 2π(2) ··· nπ(n),whe e σ(π) is a signa u e o π.
We ob ain ha he g oup o au omo phisms o Eis {Tπ|π∈Sn}and Tπ◦Tτ=Tτ◦π.
Le us ix one π∈Sn.Then T(ei) = i,π(i)eπ(i) o all 1 ≤i≤n. Now
T(ei)·T(ei) = 2
i,π(i)(eπ(i)·eπ(i)) = 2
i,π(i)
n
X
k=1
aπ(i)kek=
2
i,π(i)
n
X
k=1
aπ(i)π(k)eπ(k)=
n
X
k=1
2
i,π(i)
k,π(k)
aπ(i)π(k)T(ek).
Hence, he elemen s o e olu ion ma ix B= (bij)i,j=1,n o isomo phic algeb a o E
sa is y (3.1).

6 L. M. CAMACHO, J. R. G ´
OMEZ, B. A. OMIROV, R. M. TURDIBAEV
Fo a π∈Sndeno e by sπ:{1,2,...,n} {π−1(n)} → {1,2,...,n}a one- o-one
mapping de ined by sπ(i) = π−1(1 + π(i)).
P oposi ion 3.2. Le A= (aij)1≤i,j≤nbe a ma ix o an e olu ion algeb a isomo phic
o an e olu ion algeb a wi h Jo dan cell ma ix wi h non-ze o eigen alue λ. Then he
only non-ze o elemen s o Aa e he diagonal elemen s and ai,sπ(i) o all i6=π−1(n)
and λ=a2
ii
ai,sπ(i)asπ(i)sπ(i)
o all i6=π−1(n).
P oo . Fi s conside he isomo phism o e olu ion algeb a wi h Jo dan cell ma ix wi h
non-ze o eigen alue λ. Since he ma ix is non-singula , by he p oo o P oposi ion 3.1
we ob ain ha i is in he o m Tπ.
Fo ixed π∈Snwe pu Tπ(ei) = iand de i e
i· i= i,π(i)λ i+ 2
i,π(i)
sπ(i),π(sπ(i))
sπ(i).
Hence he ma ix o he new e olu ion algeb a is a sum o non-singula diagonal
ma ix and a ma ix ha has exac ly one non-ze o elemen on each ow excep he
π−1(n)− h, which is a ze o ow and a mos one non-ze o elemen in each column.
Now le us ix a pe mu a ion π∈Snand conside ma ix A= (aij)n
i,j=1 wi h ze o
elemen s excep he diagonal elemen s and ai,sπ(i) o all i6=π−1(n) and sπ(i) = π−1(1+
π(i)).I his e olu ion algeb a is isomo phic o an e olu ion algeb a wi h Jo dan cell
ma ix wi h eigen alue λ hen aii =λ i,π(i) o all 1 ≤i≤nand ai,sπ(i)= 2
i,π(i)
sπ(i),π(sπ(i)) .
Since i,π(i)=1
λaii and sπ(i),π(sπ(i)) =1
λasπ(i)sπ(i)we ob ain
ai,sπ(i)=a2
ii
λ2·λ
asπ(i)sπ(i)
=1
λ·a2
ii
asπ(i)sπ(i)
and hence λ=a2
ii
ai,sπ(i)asπ(i)sπ(i)
.
Hence, i ma ix Asa is ies λ=a2
ii
ai,sπ(i)asπ(i)sπ(i) o all i6=π−1(n), hen e olu ion
algeb a wi h ma ix Ais isomo phic o e olu ion algeb a wi h Jo dan cell ma ix wi h
eigen alue λ. This isomo phism has he ma ix which is he in e se o T= ( ij)n
i,j=1,
whe e iπ(i)=1
λaii and ze o o he wise. 
The abo e esul can be gene alized o he case o Jo dan o m ma ices. Le J=
J1⊕J2⊕ · · · ⊕ J ,whe e Jia e Jo dan cells o dimension niwi h non-ze o eigen alue
λi.
Now le us deno e µk=λi o n1+···+ni−1+ 1 ≤k≤n1+···+ni,1≤i≤ .
Take π∈Snand deno e s′
π:{1,...,n} {π−1(n1),...,π−1(n )} → {1,...,n}a
one- o-one mapping de ined by s′
π(i) = π−1(1 + π(i)).
Co olla y 3.3. Le A= (aij )1≤i,j≤nbe a ma ix o an e olu ion algeb a isomo phic o
an e olu ion algeb a wi h Jo dan o m ma ix J. Then he only non-ze o elemen s o A
a e he diagonal elemen s and ai,s′
π(i)such ha a2
ii
ai,s′
π(i)as′
π(i)s′
π(i)
=µ2
i
µs′
π(i)
o
i6∈ {π−1(n1),...,π−1(n )}.
4. Nilpo ency o e olu ion algeb as
Le us now conside an e olu ion algeb a Ewi h Jo dan cell wi h eigen alue λ.
P oposi ion 4.1. I λ6= 0 hen Eis nei he sol able no igh nilpo en and he e o e
is no nilpo en .
SOME PROPERTIES OF EVOLUTION ALGEBRAS 7
P oo . Since λ6= 0 hen e olu ion ma ix is non-degene a ed. The e o e, E2=E(2) =
E<2>=E. By simple induc ion we ob ain Ek=E(k)=E<k> =Eand he s a emen
o he p oposi ion is e i ied. 
P oposi ion 4.2. Fo an e olu ion algeb a wi h Jo dan cell ma ix and eigen alue
λ= 0 he ollowing s a emen s hold:
(i) Eis one gene a ed;
(ii) Eis sol able wi h index o sol abili y n+ 1;
(iii) Eis igh nilpo en wi h index o igh nilpo ency n+ 1;
(i ) Eis nilpo en wi h index o nilpo ency 2n−1+ 1.
P oo . (i) F om λ= 0 i ollows ha o basis elemen s eio Ewe ha e [ei, ei] = ei+1
o all 1 ≤i≤n−1 and [en, en] = 0.
The e o e, Eis one-gene a ed: E=idhe1i.
(ii) Fi s obse e ha E(2) =he2,...,eni.
I o some kwe ha e
E(k)=hek, ek+1,...,eni,
hen o k+ 1 we ob ain
E(k+1) =E(k)E(k)=hek+1,...,eni.
The e o e, E(n)=heniand E(n+1) = 0 and (ii) is e i ied.
(iii) is simila o (ii).
(i ) We claim ha
E2k+1 =E2k+2 =···=E2k+1 =hek+2,...,eni
o all 0 ≤k≤n−2.
Indeed, o k= 0 we ha e E2=EE =he2,...,eni.
Fo k= 1 we ha e
E2+1 =E3=EE2=he3,...,eni,
E22=E4=EE3+E2E2=he3,...,eni.
Assume ha
E2k−1+1 =E2k−1+2 =···=E2k=hek+1,...,eni.
Using his assump ion we ob ain
E2k+1 =EE2k+E2E2k−1+···+E2k−1E2k−1+1 =
EE2k+E2E2k+···+E2k−1E2k=
(E+E2+E3+···+E2k−1)E2k=EE2k=hek+2,...,eni.
Also
E2k+1 =EE2k+1−1+E2E2k+1−2+···+E2kE2k⊇
E2kE2k=hek+2,...,eni.
So we ob ain
hek+2,...,eni=E2k+1 ⊇E2k+2 ⊇ · · · ⊇ E2k+1 ⊇ hek+2,...,eni.
Hence,
E2k+1 =E2k+2 =···=E2k+1 =hek+2,...,eni.
8 L. M. CAMACHO, J. R. G ´
OMEZ, B. A. OMIROV, R. M. TURDIBAEV
The e o e, E2n−1=heniand E2n−1+1 = 0.
Hence, Eis nilpo en wi h nilpo ency index equal o 1 + 2n−1.
Rema k 4.3. We should no e ha he s a emen s (ii)−(i )o P oposi ion 4.2 a e
equi alen , since one can show ha each o hem is equi alen o λ= 0.Howe e ,
s a emen (i)is no equi alen o λ= 0 since o λ= 1 one can p o e ha Eis
gene a ed by he elemen e1+e2.
Obse e ha any e olu ion subalgeb a o an e olu ion algeb a is an ideal. The e o e i
we conside an e olu ion algeb a EJwi h ma ix Jin Jo dan o m J=J1⊕J2⊕· · ·⊕J
whe e Jia e Jo dan cells o dimension niwi h eigen alues λi, hen
EJ=E1⊕E2⊕ · · · ⊕ E ,
whe e Ei=heni−1+1,...,enii.
Now we ha e Ek
J=Ek
1⊕Ek
2⊕ · · · ⊕ Ek
and he e o e EJis nilpo en i and only
i e e y Eiis nilpo en . Since we ha e ob ained he c i e ia o nilpo ency o Jo dan
blocks, we ob ain
Co olla y 4.4. EJis nilpo en (wi h index o nilpo ency equal o max1≤i≤ {1 +2ni−1})
i and only i Jhas only ze o eigen alues. The same asse ion holds o igh nilpo ency
and sol abili y wi h co esponding indexes equal o 1 + max1≤i≤ {ni}.
No e ha om he Co olla y 4.4 i ollows ha o e e y 1 ≤k≤nwe ob ain an
example o nilpo en e olu ion algeb a wi h index o nilpo ency equal o 1 + 2k−1.
The ollowing heo em ep esen s he c i e ia o nilpo ency o ini e dimensional e o-
lu ion algeb a.
Theo em 4.5. Le Ebe an n−dimensional e olu ion algeb a. Then Eis nilpo en i
and only i he ma ix o e olu ion algeb a Acan be ans o med by he na u al basis
pe mu a ion o o m (2.3). Mo eo e , he index o nilpo ency o e olu ion algeb a Eis
no g ea e hen 2n−1+ 1.
P oo . Le Ebe a nilpo en . Then i is igh nilpo en and he e o e, by Theo em 2.7 he
ma ix o his e olu ion algeb a can be ans o med by he na u al basis pe mu a ion
o om (2.3).
Now le he ma ix Ao Ecan be ans o med by he na u al basis pe mu a ion o
o m (2.3).
Assume ha a12a23 . . . an−1n6= 0.Simila o he p oo o (i ) in P oposi ion 4.2 one
can e i y
E2k+1 =E2k+2 =···=E2k+1 =hek+2,...,eni
o all 0 ≤k≤n−2.
The e o e, E2n−1=heniand E2n−1+1 = 0.
Hence, Eis nilpo en wi h nilpo ency index equal o 1 + 2n−1.
Now assume ha a12a23 ...an−1n= 0.In his case we claim ha
hek+2,...,eni ⊇ E2k+1
o all 0 ≤k≤n−2.
Indeed, o k= 0 we ha e E2=EE ⊆ he2,...,eni.
Fo k= 1 we ha e
E2+1 =E3=EE2⊆ he3,...,eni.
SOME PROPERTIES OF EVOLUTION ALGEBRAS 9
Assume ha
hek+1,...,eni ⊇ E2k−1+1.
Using his assump ion we ob ain
E2k+1 =EE2k+E2E2k−1+···+E2k−1E2k−1+1 ⊆
EE2k−1+1 +E2E2k−1+1 +···+E2k−1E2k−1+1 =
(E+E2+E3+···+E2k−1)E2k−1+1 ⊆EE2k−1+1 ⊆ hek+2,...,eni.
So we ob ain
hek+2,...,eni ⊇ E2k+1.
The e o e, heni ⊇ E2n−2+1.
Hence,
E2n−1+1 =EE2n−1+E2E2n−1−1···+E2n−2E2n−2+1 ⊆
(E+E2+. . . E2n−2)heni ⊆ Eheni= 0.
Thus, Eis nilpo en wi h nilpo ency index no g ea e hen 1 + 2n−1.
Co olla y 4.6. Fo ini e dimensional complex e olu ion algeb a no ions as nil, nilpo-
en and igh nilpo en algeb as a e equi alen . Howe e , he indexes o nili y, igh
nilpo ency and nilpo ency do no coincide in gene al.
The ollowing p oposi ion excludes signi ican ly many possible alues ha a nilpo-
ency indexes o n−dimensional e olu ion algeb as can ake.
P oposi ion 4.7. Le Ebe a nilpo en e olu ion algeb a wi h index o nilpo ency no
equal o 2n−1+ 1.Then i is no g ea e hen 2n−2+ 1.
P oo . Since Eis nilpo en , we assume ha he ma ix Ao Ein he na u al basis
{e1,...,en}is in he o m (2.3).
F om he p oo o P oposi ion 4.5 i ollows ha a12a23 . . . an−1n= 0 and
hek+2,...,eni ⊇ E2k+1
o all 0 ≤k≤n−2.
Assume ha Eis nilpo en wi h index o nilpo ency g ea e hen 2n−2+ 1 and no
equal o 2n−1+ 1.
Then heni ⊇ E2n−2+1 and since E2n−2+1 6= 0 we ob ain E2n−2+1 =heni.
The e o e, hen−1, eni ⊇ E2n−3+1 ⊇E2n−3+2 ⊇ · · · ⊇ E2n−2⊇ heni.
Now i E2n−3+1 =E2n−3+2 =···=E2n−2=heni hen
E2n−2+1 =EE2n−2+E2E2n−2−1···+E2n−3E2n−3+1 ⊆
(E+E2+···+E2n−3)heni=Eheni= 0
which is a con adic ion. Hence, hen−1, eni=E2n−3+1.
Now assume ha hen−k,...,eni=E2n−k−2+1.
Then
hen−k−1, en−k,...,eni ⊇ E2n−k−3+1 ⊇E2n−k−3+2 ⊇ · · · ⊇ E2n−k−2⊇ hen−k,...,eni.
I E2n−k−3+1 6=hen−k−1, en−k,...,eni hen E2n−k−3+1 =E2n−k−3+2 =··· =E2n−k−2=
hen−k,...,eniand
E2n−k−2+1 =EE2n−k−2+···+E2n−k−3E2n−k−3+1 =Ehen−k, en−k,...,eni ⊆ hen−k+1, en−k,...,eni
which con adic s o hen−k,...,eni=E2n−k−2+1.