On abelian subalgebras and ideals of maximal dimension in supersolvable Lie algebras

a Xi :1110.2389 3 [ma h.RA] 5 Jun 2013 ON ABELIAN SUBALGEBRAS AND IDEALS OF MAXIMAL DIMENSION IN SUPERSOLVABLE LIE ALGEBRAS Manuel Ceballos 1 Depa men o de Geome ia y Topologia, Uni e sidad de Se illa Apa ado 1160, 41080, Se ille, Spain and Da id A. Towe s Depa men o Ma hema ics, Lancas e Uni e si y Lancas e LA1 4YF, England Abs ac In his pape , he main objec i e is o compa e he abelian subalge- b as and ideals o maximal dimension o ini e-dimensional supe sol - able Lie algeb as. We cha ac e ise he maximal abelian subalgeb as o sol able Lie algeb as and s udy sol able Lie algeb as con aining an abelian subalgeb a o codimension 2. Finally, we p o e ha nilpo en Lie algeb as wi h an abelian subalgeb a o codimension 3 con ain an abelian ideal wi h he same dimension, p o ided ha he cha ac e is ic o he unde lying ield is no wo. Th oughou he pape , we also gi e se e al examples o cla i y some esul s. Ma hema ics Subjec Classi ica ion 2010: 17B05, 17B20, 17B30, 17B50. Key Wo ds and Ph ases: Lie algeb as, abelian subalgeb a, abelian ideal, sol able, supe sol able, nilpo en . 1 In oduc ion Nowadays, he e exis s an ex ensi e body o esea ch o Lie Theo y due o i s own impo ance om a heo e ical poin o iew and also due o i s 1Suppo ed by MTM2010-19336 and FEDER 1 applica ions o o he ields like Enginee ing, Physics and Applied Ma he- ma ics. Howe e , some aspec s o Lie algeb as emain unknown. Indeed, he classi ica ion o nilpo en and sol able Lie algeb as is s ill an open p ob- lem, al hough he classi ica ion o ce ain o he ypes o Lie algeb as (like semi-simple and simple ones) we e al eady ob ained in 1890, a leas o e he complex ield. In o de o make p og ess on hese and o he p oblems, he need o s udying di e en p ope ies o Lie algeb as a ises. Fo exam- ple, condi ions on he la ice o subalgeb as o a Lie algeb a o en lead o in o ma ion abou he Lie algeb a i sel . S udying abelian subalgeb as and ideals o a ini e-dimensional Lie algeb a cons i u es he main goal o his pape . Th oughou Lwill deno e a ini e-dimensional Lie algeb a o e a ield F. The assump ions on Fwill be speci ied in each esul . Algeb a di ec sums will be deno ed by ⊕, whe eas ec o space di ec sums will be deno ed by ˙ +. We conside he ollowing in a ian s o L: α(L) = max{dim(A)|Ais an abelian subalgeb a o L}, β(L) = max{dim(B)|Bis an abelian ideal o L}. Bo h in a ian s a e impo an o many easons. Fo example, hey a e e y use ul o he s udy o Lie algeb a con ac ions and degene a ions. The e is a la ge li e a u e, in pa icula o low-dimensional Lie algeb as, see [9, 6, 13, 15, 8], and he e e ences gi en he ein. The i s au ho dealing wi h he in a ian α(g) was Schu [14], who s udied in 1905 he abelian subalgeb as o maximal dimension con ained in he Lie algeb a o n×nsqua e ma ices. Schu p o ed ha he max- imum numbe o linea ly independen commu ing n×nma ices o e an algeb aically closed ield is hn2 4i+ 1, which is he maximal dimension o abelian ideals o Bo el subalgeb as in he gene al linea Lie algeb a gl(n) ( whe e [x] deno es he in ege pa o a eal numbe x). Le us no e ha his esul was ob ained only o e an algeb aically closed ield such as he complex numbe ield. Almos o y yea s la e , in 1944, Jacobson [10] ga e a simple p oo o Schu ’s esul s, ex ending hem om algeb aically closed ields o a bi a y ields. This ac allowed se e al au ho s o gain insigh in o he abelian subalgeb as o maximal dimension o many di e en ypes o Lie algeb as. Mo e speci ically, o semisimple Lie algeb as s he in a ian α(s) has been comple ely de e mined by Malce [12]. Since he e a e no abelian ideals in s, we ha e β(s) = 0. The alue o α o simple Lie algeb as is ep oduced in able 1. In his pape , we will s udy se e al p ope ies o 2 hese in a ian s and compa e hem o supe sol able, sol able and nilpo en Lie algeb as. Table 1: The in a ian α o simple Lie algeb as sdim(s)α(s) An, n ≥1n(n+ 2) ⌊(n+1 2)2⌋ B321 5 Bn, n ≥4n(2n+ 1) n(n−1) 2+ 1 Cn, n ≥2n(2n+ 1) n(n+1) 2 Dn, n ≥4n(2n−1) n(n−1) 2 G214 3 F452 9 E678 16 E7133 27 E8248 36 We shall call Lsupe sol able i he e is a chain 0 = L0⊂L1⊂... ⊂ Ln−1⊂Ln=L, whe e Liis an i-dimensional ideal o L. The ideals L(k)o he de i ed se ies a e de ined by L(0) =L, L(k+1) = [L(k), L(k)] o k≥0; we also w i e L2 o L(1) and L3 o [L2, L]. I is well known ha e e y supe sol able Lie algeb a is also sol able. Mo eo e , hese classes coincide o e an algeb aically closed ield o cha ac e is ic ze o (Lie’s heo em). The e a e, howe e , examples o sol able Lie algeb as o e algeb aically closed ield o non-ze o cha ac e is ic which a e no supe so able (see o ins ance [11, page 53] o [3]). The F a ini ideal o L,φ(L), is he la ges ideal o L con ained in all maximal subalgeb as o L. We will deno e he cen e o L by Z(L) = {x∈L: [x, y] = 0,∀y∈L}and he cen alize o a subalgeb a Ao Lby CL(A) = {x∈L: [x, A] = 0}. Gi en a subalgeb a Ao L, he co e o A, deno ed by AL, is he la ges ideal o Lcon ained in A. The abelian socle o L, AsocL, is he sum o he minimal abelian ideals o L. The s uc u e o his pape is as ollows. In sec ion 2 we gi e some bounds o he in a ian s αand β. In sec ion 3, we conside he classes o supe sol able, sol able and nilpo en Lie algeb as Lwi h α(L) = n−1 o n−2. In pa icula , we cha ac e ise n-dimensional sol able Lie algeb as L o which α(L) = n−2 and p o e ha e e y supe sol able Lie algeb a, L, o dimension nwi h α(L) = n−2 also sa is ies β(L) = n−2. In he inal sec ion we show he αand βin a ian s also coincide o nilpo en Lie 3 algeb as Lwi h α(L) = n−3, p o ided ha Fhas cha ac e is ic di e en om wo. We also gi e an example o show ha he es ic ion on Fis necessa y. 2 Some bounds on α(L)and β(L) We shall call a Lme abelian i L2is abelian. Fi s we ha e a bound on β(L) o ce ain me abelian Lie algeb as. P oposi ion 2.1 Le Lbe a me abelian Lie algeb a o dimension n, and suppose ha dim L2=k. Then dim(L/CL(L2)) ≤[k2/4] + 1. I , u he , L spli s o e L2 hen β(L)≥n−[k2/4] −1. P oo . Le ad : L→De L2be de ined by ad x(y) = [y, x] o all y∈L2. Then ad is a homomo phism wi h ke nel CL(L2). I ollows ha L/CL(L2)∼ = Dwhe e Dis an abelian subalgeb a o De L2∼ =gl(n, F ). I ollows om Schu ’s Theo em on commu ing ma ices (see [10]) ha dim(L/CL(L2)≤ [k2/4] + 1. Now suppose ha L=L2⊕Bwhe e Bis an abelian subalgeb a o L. Then CL(L2) = L2⊕B∩CL(L2) which is an abelian ideal o L. We call Lcomple ely sol able i L2is nilpo en . O e a ield o cha ac- e is ic ze o, e e y sol able Lie algeb a is comple ely sol able. Nex we no e ha i Lis comple ely sol able, has an abelian nil adical (so is me abelian) and he unde lying ield is pe ec hen α(L) and β(L) a e easily iden i ied. I ¯ Fis he algeb aic closu e o Fwe pu ¯ S=S⊗F¯ F o e e y subalgeb a So L. Lemma 2.2 α(¯ L)≥α(L),β(¯ L)≥β(L). Lemma 2.3 Le Lbe any sol able Lie algeb a wi h nil adical N. Then CL(N)⊆N P oo . Suppose ha CL(N)6⊆ N. Then he e is a non- i ial abelian ideal A/(N∩CL(N) o L/(N∩CL(N) inside CL(N)/(N∩CL(N). Bu now A3⊆ [A, N] = 0, so Ais a nilpo en ideal o L. I ollows ha A⊆N∩CL(N), a con adic ion.  Theo em 2.4 I Fis a pe ec ield and Lis a comple ely sol able Lie algeb a wi h abelian nil adical N hen α(L) = β(L) = dim N. 4 P oo . I is clea ha Nis he unique maximal abelian ideal o L. Le Abe an abelian subalgeb a o Lo maximal dimension. I N⊆A, hen A⊆CL(N) = N, by Lemma 2.3, so N=Aand he esul is clea , so suppose ha N6⊆ Aand pu U=N+A. Conside i s he case whe e Fis algeb aically closed and φ(L) = 0. Pick any a∈Aand pu C=N+Fa. Then φ(C) = 0, by [18, Theo em 2.5], so N⊆N(C) = Asoc(C) by [17, Theo em 7.4], and Nis comple ely educible as an Fa-module. W i e N=⊕k i=1Ni, whe e Niis an i educible Fa-module o 1 ≤i≤k. Then he minimal polynomial o he es ic ion o ad a o Niis i educible o each i, and so {(ada)|N:a∈A}is a se o commu ing diagonalizable ope a o s. Thus N= AsocL=Fn1+. . . Fn , whe e Fniis a minimal ideal o L o 1 ≤i≤ . I ni∈A, hen CU(ni) = U; i ni/∈A hen dim CU(ni)≥dim U−1, since U/CU(ni)∼ =D, whe e Dis a subalgeb a o De Fni. Bu N=CL(N)⊇CU(N) = i=1 CU(Fni), so dim N≥dim U−( −dim(N∩A)) = dim A, and he esul holds in his case. So suppose now ha φ(L) is no necessa ily i ial. Then N/φ(L) is he nil adical o L/φ(L), by [17, Theo em 6.1], and so L/φ(L) has abelian nil adical. Also (A+φ(L))/φ(L) is an abelian subalgeb a o L/φ(L). I ollows om he abo e ha dim A+φ(L) φ(L)≤dim N φ(L),whence dim A≤dim N. Finally conside he case whe e Fis no necessa ily algeb aically closed. Since Fis pe ec , N(L) = N(¯ L), by [4, page 42]. Hence β(L)≤α(L)≤α(¯ L) = β(¯ L) = dim N(¯ L) = dim N(L) = dim N(L) = β(L) by Lemma 2.2 and he abo e.  We ob ain bounds o supe sol able Lie algeb as by ollowing a de elop- men simila o [16, Lemma 2]. Lemma 2.5 Le Lbe a supe sol able Lie algeb a and le Abe a maximal abelian ideal o L. Then CL(A) = A. 5 P oo . We ha e ha CL(A) is an ideal o L. Suppose ha CL(A)6=A. Le B/A be a minimal ideal o L/A wi h B⊂CL(A). Then, o some b∈B, B=A+Fb, which is an abelian ideal o L, con adic ing he maximali y o A. The esul ollows.  P oposi ion 2.6 Le Lbe a supe sol able Lie algeb a, Aany maximal abelian ideal o L. Suppose dim A=k. Then L/A is isomo phic o a Lie algeb a o k×klowe iangula ma ices. P oo . Le ad : L→De Abe de ined by ad x(y) = [y, x]. Then ad is a homomo phism wi h ke nel CL(A) = A, by Lemma 2.5. Since Lis supe sol able he e is a lag o ideals 0 = A0⊂A1⊂... ⊂Ak=Ao L. Choose a basis e1,...,ek o Awi h ei∈Ai. Wi h espec o his basis he ac ion o Lon Ais ep esen ed by k×klowe iangula ma ices, since [Ai, L]⊆Ai o each 0 ≤i≤k. Co olla y 2.7 Le Lbe a supe sol able Lie algeb a wi h a maximal abelian ideal Ao dimension k. Then dim L≤k(k+3) 2and Lhas de i ed leng h a mos k+ 1. P oo . The Lie algeb a o k×klowe iangula ma ices has dimension k(k+1) 2and de i ed leng h k. Co olla y 2.8 Le Lbe a supe sol able Lie algeb a o dimension n. Then β(L)≥√8n+ 9 −3 2. Co olla y 2.9 Le Lbe a non-abelian sol able Lie algeb a o dimension n o e an algeb aically closed ield o cha ac e is ic ze o. Then √8n+ 9 −3 2≤α(L)≤n−1. P oo . Simply use Co olla y 2.8 and [7, P oposi ion 2.5].  3 Supe sol able Lie algeb as wi h α(L) = n−1o n−2 P oposi ion 3.1 Le Abe an abelian subalgeb a o a Lie algeb a L. Suppose ha K=A+Fe1is a subalgeb a o L, and ha he e is an x∈Lsuch ha [x, K]⊆K, bu [x, A]6⊆ A. Then ei he Kis abelian o K2is one dimensional and Z(K)has codimension a mos one in A. 6 P oo . Le e2,...,ekbe a basis o Asuch ha e1= [x, e2], say. Le [x, ej] = Pk i=1 αjiei o 1 ≤j≤k. Then [e2,[x, ej]] = αj1[e2, e1], so, o 2≤j≤k, 0 = [x, [e2, ej]] = −[e2,[ej, x]] −[ej,[x, e2]] = αj1[e2, e1]−[ej, e1]. Hence [e1, ej] = αj1[e1, e2]. I ollows ha K2=F[e1, e2]. Pu j=αj1e2− ej o 3 ≤j≤k. Then 3,..., k∈Z(K)∩A. The abo e esul deals wi h he case whe e an abelian subalgeb a o maximal dimension has codimension one in an ideal o L. Co olla y 3.2 Le Lbe a supe sol able Lie algeb a and le Abe an abelian subalgeb a o maximal dimension in L. I A⊂Kwhe e Kis an ideal o L and Ahas codimension 1in K, hen α(L) = β(L). P oo . I Ais an ideal o L hen he esul is clea , so suppose ha i is no an ideal o L. Wi h he same no a ion as in P oposi ion 3.1 he hypo heses o ha esul a e sa is ied. Then 3,..., k∈Z(K); in ac , he maximali y o Agi es Z(K) = F 3+···+F k. Le B/(F 3+···+F k) be a chie ac o o Lwi h B⊂K. Then Bis an abelian ideal o Lwi h he same dimension as A. The esul ollows.  Nex we conside he si ua ion whe e Lhas a maximal subalgeb a ha is abelian: i s when Lis any non-abelian Lie algeb a and Fis algeb aically closed, and hen when Lis sol able bu Fis a bi a y. P oposi ion 3.3 Le Lbe a non-abelian Lie algeb a o dimension no e an algeb aically closed ield Fo any cha ac e is ic. Then Lhas a maximal subalgeb a M ha is abelian i and only i L=A˙ +F o some ∈gl(V), whe e 6≡ 0,Ais abelian and [ , ] = −[ , ] = ( ). In pa icula , α(L) = β(L) = n−1. P oo . Suppose i s ha Lhas a maximal subalgeb a M ha is abelian. I Mis an ideal o Lwe ha e inished. So suppose ha Mis sel -idealising, in which case L2is one-dimensional and φ(L) = 0, by [19, P oposi ion 3.2]. W i e L2=Fb and no e ha AsocL=L2⊕Z(L). Now L= AsocL˙ +C, whe e Cis abelian, by [17, Theo em 7.4]. Fo each c∈Cwe ha e ha [c, b] = λ(c)b, o some λ(c)∈F. Since C∩Z(L) = 0, Cmus be one- dimensional and he esul ollows. The con e se is clea .  The ollowing is a gene alisa ion o [19, P oposi ion 3.1] 7 P oposi ion 3.4 Le Lbe a sol able Lie algeb a. Then Lhas a maximal subalgeb a M ha is abelian i and only i ei he (i) Lhas an abelian ideal o codimension one in L; o (ii) L(2) =φ(L) = Z(L),L2/L(2) is a chie ac o o L, and Lspli s o e L2. P oo . By [19, P oposi ion 3.1] i su ices o show ha i Lhas a maximal subalgeb a M ha is no an ideal hen L2is nilpo en . By maximali y, M is sel -idealising, and om sol abili y he e is a k≥1 such ha L(k)6⊆ M bu L(k+1) ⊆M. Then L=M+L(k), whence L2⊆L(k). I ollows ha L(2) ⊆Mand we ha e M⊆CL(L(2)). By maximali y and he ac ha M is sel -idealising, we ge CL(L(2)) = L, which yields ha L2is nilpo en .  Nex we cha ac e ise sol able Lie algeb as Lwhose bigges abelian subal- geb as ha e codimension wo in L. The ollowing p oo elies on [7, P opo- si ions 3.1 and 5.1] which a e only s a ed o Lie algeb as o e ields o cha ac e is ic ze o. Howe e , i is easy o see ha his assunp ion is no used in hei p oo s, and ha he esul s a e, in ac , alid o e an a bi a y ield. Theo em 3.5 Le Lbe a sol able Lie algeb a o dimension nwi h α(L) = n−2, and le Abe an abelian subalgeb a o dimension n−2. Then one o he ollowing occu s: (i) β(L) = n−2; (ii) L=L2˙ +B, whe e Bis an abelian subalgeb a o L,L2is he h ee- dimensional Heisenbe g algeb a, L(2) =φ(L) = Z(L)and L2/Z(L)is a wo-dimensional chie ac o o L(in which case β(L)≤n−3); (iii) Ahas codimension one in he nil adical, N, o L, which i sel has codimension one in L. Mo eo e , N2is one dimensional, Z(N)is an abelian ideal o maximal dimension and β(L) = n−3. P oo . Le Abe a maximal abelian subalgeb a o Lo dimension n−2 and suppose ha (i) doesn’ hold. (a) Suppose i s ha Ais a maximal subalgeb a o L. Then Lis as in P oposi ion 3.4(ii) and Ais a Ca an subalgeb a o L. Le L=A˙ +L1 be he Fi ing decomposi ion o L ela i e o A. Then L1⊆L2and dim L1= 2. Le L1=Fx +Fy. 8 I [x, y] = 0 hen L1is an ideal o Land L/L1is abelian, so L2⊆ L1⊆L2. This yields ha Lis me abelian and L2is a wo dimensional minimal ideal o e which Lspli s. I ollows om P oposi ion 2.1 ha β(L) = n−2, a con adic ion. I [x, y]6= 0, hen L2=F[x, y] + L1and F[x, y]⊆L(2) =Z(L), so F[x, y] = Z(L) and we ha e case (ii). Mo eo e , i Cis a maximal abelian ideal o L, hen Z(L)⊆Cand L26⊆ C. I ollows ha C∩L2=Z(L). I dim C=n−2, hen dim(L2+C) = dim L2+ dim C−dim C∩L2= 3 + n−2−1 = n, so L=L2+C. Bu hen L2=Z(L), a con adic ion. Hence, β(L)≤n−3. (b) So suppose ha Ais no a maximal subalgeb a o L. Then A⊂M⊂L, whe e dim M=n−1. Mo eo e , he e is such a subalgeb a Ao M which is an ideal o M, by [7, P oposi ion 3.1]. Suppose i s ha A does no ac nilpo en ly on L. Then he Fi ing decomposi ion o L ela i e o Ais L=M˙ +L1, and L1is a one-dimensional ideal o L. Pu B=A˙ +L1, which is an ideal o L. Then CB(L1) has codimension one in Band so is an abelian ideal o codimension wo in L. I ollows ha β(L) = n−2, a con adic ion. Finally, suppose ha Ais an ideal o Mand ha Aac s nilpo en ly on L. Then he e is a k≥0 such ha L(ad A)k6⊆ Mbu L(ad A)k+1 ⊆ M. Le x∈L(ad A)k M, so L=M˙ +Fx. Suppose i s ha Mis no an ideal o L. Then he co e o M,MLhas codimension one in M, by [1, Theo em 3.1 and 3.2]. I A=ML hen we ha e case (i), so suppose ha A6=MLand M=A+ML. Then [A, x]⊆Mwhich implies ha [L, A]⊆Mand [L, M] = [L, ML] + [L, A]⊆M; ha is, Mis an ideal o L. Le Nbe he nil adical o L. I N⊆A hen A⊆CL(N)⊆N, so N=Aand we ha e case (i) again. I A⊂N hen N=Lo we can assume ha N=M. I A6⊆ Nand N6⊆ A hen ei he A+N=L, in which case Lis nilpo en , o we can assume ha A+N=M, in which case Mis a nilpo en ideal o Land so M=N. I Lis nilpo en , hen we ha e case (i), by [7, P oposi ion 5.1].I no , hen we ha e case (iii) by P oposi ion 3.1.  Co olla y 3.6 Le Lbe a supe sol able Lie algeb a o dimension nwi h α(L) = n−2. Then β(L) = n−2. 9 [18] D. A. Towe s and V. R. Va ea, ‘Elemen a y Lie Algeb as and Lie A-algeb as’, J. Algeb a 312 (2007), 891–901. [19] D. A. Towe s, ‘The index complex o a maximal subalgeb a o a Lie algeb a’, P oc. Edin. Ma h. Soc. 54 (2011), 531–542. 16