We study the problem of the existence of filtered multiplicative bases of a restricted enveloping algebra u(L), where L is a finite-dimensional and p-nilpotent restricted Lie

algebra over a field of positive characteristic p

Alexander Grishkov

AV Roiter

DM Riley

H Kupisch

H Strade

J Feldvoss

M Lincoln

P Gabriel

P Landrock

R Bautista

S Siciliano

Salvatore Siciliano

V Bovdi

Victor Bovdi

Y Bahturin

Z Balogh

arXiv

We study the problem of the existence of filtered multiplicative
bases of a restricted enveloping algebra u(L), where L is a finite-dimensional and p-nilpotent restricted Lie algebra over a field of positive characteristic p

Victor BOVDI

Alexander GRISHKOV

SICILIANO, Salvatore

Archivio Istituzionale della Ricerca- Università del Salento

Filtered multiplicative bases of restricted enveloping algebras

We study the problem of the existence of filtered multiplicative bases of a restricted enveloping algebra u(L), where L is a finite-dimensional and p-nilpotent restricted Lie algebra over a field of positive characteristic p.OTKA[K68383]OTKARFFI[07-01-00392A]RFFIFundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)FAPESPCNPq(Brazil)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq

BOVDI, Victor

GRISHKOV, Alexander

Universidade de São Paulo

Algebras and Representation Theory

Filtered Multiplicative Bases of Restricted Enveloping Algebras

We study the problem of the existence of filtered multiplicative bases of a restricted enveloping algebra u(L), where L is a finite-dimensional and p-nilpotent restricted Lie algebra over a field of positive characteristic p

RCAAP - Repositório Científico de Acesso Aberto de Portugal

Crossref

Biblioteca Digital da Produção Intelectual da Universidade de São Paulo (BDPI/USP)

Bódi, Viktor

Grishkov, A.

Siciliano, S.

Repository of the Academy's Library

Algebr Represent TheorDOI 10.1007/s10468-009-9203-0Filtered Multiplicative Bases of RestrictedEnveloping AlgebrasVictor Bovdi · Alexander Grishkov ·Salvatore SicilianoReceived: 9 April 2009 / Accepted: 29 November 2009© Springer Science+Business Media B.V. 2009Abstract We study the problem of the existence of filtered multiplicative bases of arestricted enveloping algebra u(L), where L is a finite-dimensional and p-nilpotentrestricted Lie algebra over a field of positive characteristic p.Keywords Filtered multiplicative basis · Restricted enveloping algebraMathematics Subject Classifications (2000) Primary 16S30 · 17B501 Introduction and ResultsLet A be a finite-dimensional associative algebra over a field F. Denote by rad(A)the Jacobson radical of A and let B be an F-basis of A. Then B is called a filteredmultiplicative basis (f.m. basis) of A if the following properties hold:(i) for every b 1, b 2 ∈ B either b 1b 2 = 0 or b 1b 2 ∈ B;Presented by Raymundo Bautista.The research was supported by OTKA No. K68383, RFFI 07-01-00392A, FAPESP andCNPq(Brazil).V. Bovdi (B)University of Debrecen, P.O. Box 12, 4010 Debrecen, Hungarye-mail: vbovdi@math.unideb.huA. GrishkovIME, USP, Rua do Matao, 1010 – Citade Universitària,CEP 05508-090, Sao Paulo, Brazile-mail: shuragri@gmail.comS. SicilianoDipartimento di Matematica “E. De Giorgi”, Università del Salento,Via Provinciale Lecce–Arnesano, 73100, Lecce, Italye-mail: salvatore.siciliano@unile.itV. Bovdi et al.(ii) B ∩ rad(A) is an F-basis of rad(A).Filtered multiplicative bases arise in the theory of representation of associativealgebras and were introduced by H. Kupisch in [10]. In their celebrated paper [4]R. Bautista, P. Gabriel, A. Roiter and L. Salmeron proved that if A has finiterepresentation type (that is, there are only finitely many isomorphism classes offinite-dimensional indecomposable A-modules) over an algebraically closed field F,then A has an f.m. basis.In [9] an analogous statement was proposed for finitely spaced modules over anaggregate. (Such modules give rise to a matrix problem in which the allowed columntransformations are determined by the module structure, the row transformationsare arbitrary, and the number of canonical matrices is finite). This statement wassubsequently proved in [14].The problem of existence of an f.m. basis in a group algebra was posed in [4] andhas been considered by several authors: see e.g. [2, 3, 5–7, 11]. In particular, it is stillan open problem whether a group algebra KG has an f.m. basis in the case when Fis a field of odd characteristic p and G is a nonabelian p-group.Apparently, not much is known about the same problem in the setting of restrictedenveloping algebras. The present paper represents a contribution in this direction.In particular, because of the analogy with the theory of finite p-groups, we confineour attention to the class Fp of finite-dimensional and p-nilpotent restricted Liealgebras over a field of positive characteristic p. Note that under this assumption,the aforementioned result in [4] can be applied only in very special cases. Indeed, forL ∈ Fp, from [8] it follows that u(L) has finite representation type if and only if L iscyclic, that is, there exists an element which generates L as a restricted subalgebra.Our main results are the following three theorems.Theorem 1 Let L ∈ Fp be an abelian restricted Lie algebra over a field F. Then u(L)has a filtered multiplicative basis if and only if L decomposes as a direct sum of cyclicrestricted subalgebras. In particular, if F is a perfect field, then u(L) has a filteredmultiplicative basis.A restricted Lie algebra L ∈ Fp is called powerful (see e.g. [15]) if p = 2 and L′ ⊆L[p]2 or p > 2 and L′ ⊆ L[p]. Here L[p]i denotes the restricted subalgebra generatedby the elements x[p]i , x ∈ L.Theorem 2 Let L ∈ Fp be a nonabelian restricted Lie algebra over a field F. If L ispowerful then u(L) does not have a filtered multiplicative basis.Theorem 3 Let L ∈ Fp be a restricted Lie algebra over a field F. If L has nilpotencyclass 2 and p > 2 then u(L) does not have a filtered multiplicative basis.An example showing that Theorem 3 fails in characteristic 2 is also provided. Fi-nally, we remark that for odd p no example of noncommutative restricted envelopingalgebra having an f.m. basis seems to be known.Filtered Multiplicative Bases of Restricted Enveloping Algebras2 PreliminariesLet A be a finite-dimensional associative algebra over a field F. If B is an f.m. basisof A then the following simple properties hold (see [5]):(F-I) B ∩ rad(A)n is an F-basis of rad(A)n for every n ≥ 1;(F-II) if u, v ∈ B\ rad(A)k and u ≡ v (mod rad(A)k) then u = v.Let L be a restricted Lie algebra over a field F of characteristic p > 0 with a p-map [p]. We denote by ω(L) the augmentation ideal of u(L), that is, the associativeideal generated by L in u(L). In [13], the dimension subalgebras of L were definedas the restricted ideals of L given byDm(L) = L ∩ ω(L)m (m ≥ 1).These subalgebras can be explicitly described as Dm(L) = ∑ip j≥m γi(L)[p]j, whereγi(L)[p]jis the restricted subalgebra of L generated by the set of pj-th powers of thei-th term of the lower central series of L. Note that D2(L) coincides with the Frattinirestricted subalgebra (L) of L (cf. [12]).It is well-known that if L is finite-dimensional and p-nilpotent then ω(L) isnilpotent. Clearly, in this case ω(L) coincides with rad(u(L)) and u(L) = F · 1 ⊕ω(L). Consequently, if u(L) has an f.m. basis B, then we can assume without lossof generality that 1 ∈ B. For each x ∈ L, the largest subscript m such that x ∈ Dm(L)is called the height of x and is denoted by ν(x). The combination of Theorems 2.1 and2.3 from [13] gives the following.Lemma 1 Let L ∈ Fp be a restricted Lie algebra over a field F, and let {xi}i∈I be anordered basis of L chosen such thatDm(L) = spanF{xi | ν(xi) ≥ m} (m ≥ 1).Then for each positive integer n the following statements hold:(i) ω(L)n = spanF{x | ν(x) ≥ n}, where x = xα1i1 · · · xαlil ,ν(x) = ∑lj=1 α jν(xi j), i1 < · · · < il and 0 ≤ α j ≤ p − 1.(ii) The set { y | ν(y) = n } is an F-basis of ω(L)n modulo ω(L)n+1.For a subset S of L we shall denote by 〈S〉p the restricted subalgebra generated byS. Moreover, if z ∈ L is p-nilpotent, the minimal positive integer n such that z[p]n = 0is called the exponent of z and denoted by e(z).3 ProofsProof of the Theorem 1 Assume first that L = ⊕ni=1〈xi〉p. Then, by the PBW Theo-rem for restricted Lie algebras (see [16], Chapter 2, Theorem 5.1), we see that u(L)is isomorphic to the truncated polynomial algebraF[X1, . . . , Xn]/(X pe(x1)1 , . . . , X pe(xn)n ).Consequently the algebra u(L) has an f.m. basis.V. Bovdi et al.Conversely, suppose that u(L) has an f.m. basis B1 with 1 ∈ B1 and put B =B1\{1}. Let n = dimF L/L[p] and B\ω(L)2 = {b 1, . . . , b n}. By Lemma 1, one hasbi = xi + hi, where xi ∈ L\L[p] and hi ∈ ω(L)2 for every i = 1, . . . , n. From [12] itfollows at once that {x1, . . . , xn} is a minimal set of generators of L as a restrictedsubalgebra. We shall prove by induction on e = max{i| L[p]i = 0} that L has a cyclicdecomposition. If e = 1, then L = ⊕ni=1〈xi〉p.Now let e > 1 and suppose that L does not decompose as a direct sum of restrictedsubalgebras. Since L = ∑ni=1〈xi〉p and the p-map is p-semilinear, note that L[p]e isjust the vector subspace generated by the pe-th powers of the generators xi havingexponent e + 1. Therefore, without loss of generality we can assume thate + 1 = e(x1) = · · · = e(xm) ≥ e(xs), (m + 1 ≤ s ≤ n)and {x[p]e1 , . . . , x[p]em } is an F-linearly independent set withspanF{x[p]e1 , . . . , x[p]em } = L[p]e.In turn, we can reindex the elements xm+1, . . . , xn so that there exists a maximalm ≤ k < n such thatH = 〈x1, . . . , xm, . . . , xk〉p =k⊕i=1〈yi〉pfor suitable y1, . . . , yk in L with yi = xi for i = 1, . . . , m. Consequently, for everys > k there exists a minimal number fs such thatxpfss =k∑i=1e(yi)−1∑j=0μ(s)i, j y[p] ji (μ(s)i, j ∈ F). (1)Denote by J the associative ideal of u(L) generated by the elements b pe1 , . . . , bpem .Clearly J ⊆ ω(L)pe ⊆ L[p]e u(L). Suppose by contradiction that J = L[p]e u(L). If r isthe maximal positive integer such that(L[p]eu(L) ∩ ω(L)r) \J = ∅,then there exists v = xpei xa11 · · · xann ∈ ω(L)r \ J such thatv ≡ b pei b a11 · · · b ann (mod ω(L)r+1).Consequentlyv − b pei b a11 · · · b ann ∈(L[p]eu(L) ∩ ω(L)r+1) \J,contradicting the definition of r. Therefore J = L[p]e u(L), which implies thatu(L)/J ∼= u(L/L[p]e). Moreover, it is easily seen that B1 ∩ J is an F-basis of J, hencethe elements bi + J with bi /∈ J form an f.m. basis of u(L)/J. Consequently, byinduction we have that L = L/L[p]e is a direct sum of restricted subalgebras.As the images of y1, . . . , yk are F-linearly independent in L/(L), from [12] itfollows that there exists a restricted subalgebra P of L with L[p]e ⊆ P such thatL = H/L[p]e ⊕ P/L[p]e .Filtered Multiplicative Bases of Restricted Enveloping AlgebrasAs a consequence, for every s > k we have xs ≡ vs + ws (mod L[p]e) with vs ∈ H andws ∈ P and, moreover, it follows from Eq. 1 that w[p] fss ∈ L[p]e . One hasvs =k∑i=1e(yi)−1∑j=0k(s)i, j y[p] ji , (k(s)i, j ∈ F). (2)Since x[p]fss ≡ v[p] fss (mod L[p]e), we conclude that μ(s)i, j ∈ F p fs provided j < e. Weclaim that for every 1 ≤ i ≤ k the coefficient μ(s)i,e is also in F p fs . Indeed, writews =∑b∈Bλb b (for suitable λb ∈ F).Then, as B is a filtered multiplicative basis of u(L), it follows thatwpfss =∑b∈Cμp fsb bp fs (3)where C is a subset of B and the μb ’s are nonzero elements of F. Moreover, sincew[p] fss ∈ L[p]e we havewpfss =m∑i=1αibpei . (4)As B is a filtered F-basis of u(L), by comparing Eqs. 3 and 4 we conclude that forevery i = 1, 2, . . . , m there exists βi ∈ F such that αi = β p fsi . At this stage, since b pei =x[p]ei for every i = 1, 2, . . . , m, the relations (1) and (2) allow to conclude that forevery 1 ≤ i ≤ k one hasμ(s)i,e =(k(s)i,e)p fs + β p fsi ∈ F pfs,as desired (here βm+1, . . . , βk = 0).Now, by Eq. 1 and the above discussion we have xpfss = zp fs for some z ∈ H.Therefore (xs − z)p fs = 0 and then the minimality of fs forces 〈xs − z〉p ∩ H = 0.This contradicts the definition of k, yielding the claim.Finally, if F is perfect, then L decomposes as a direct sum of cyclic restrictedsubalgebras (see e.g. [1], Chapter 4, Theorem in Section 3.1). The proof is done. unionsqUnlike group algebras, a commutative restricted enveloping algebra need nothave an f.m. basis. Indeed, we have the followingExample Let F be a field of positive characteristic p containing an element α whichis not a p-th root in F. Consider the abelian restricted Lie algebraLα = Fx + Fy + Fzwith x[p] = αz, y[p] = z, and z[p] = 0. Suppose that u(Lα) has an f.m. basis. ByTheorem 1, Lα is a direct sum of cyclic restricted subalgebras. Since L[p]α = 0 andL[p]2α = 0, we have Lα = 〈a〉p ⊕ 〈b〉p with e(a) = 2 and e(b) = 1.Let b = k1x + k2 y + k3z, ki ∈ F. Since α /∈ F p, we get 0 = b [p] = (kp1 α + kp2 )z, sok1 = k2 = 0 and 0 = a[p] ∈ Fz = 〈b〉p, a contradiction.V. Bovdi et al.Lemma 2 Let A be a finite-dimensional nilpotent noncommutative associative algebraover a field F. Suppose that A has a minimal set of generators {u1, . . . , un} such that:(i) [ui, u j] ∈ A3 for every i, j = 1, . . . , n;(ii) uiu j /∈ A3 for every i, j = 1, . . . , n;(iii) spanF{uiu j | 1 ≤ i < j ≤ n} ∩ spanF{u2i | i = 1, . . . , n} ⊆ A3.Then A has no f.m. basis.Proof By contradiction, assume that there exists an f.m. basis B of A. Clearly, wehave dimF A/A2 = n and, by property (F-I), B\ω(L)2 is a minimal set of generatorsof A as an associative algebra. Write B\A2 = {b 1, . . . , b n}. Obviouslyb k ≡n∑i=1αkiui (mod A2), (αki ∈ F)and the determinant of the matrix M = (aki) is not zero. Nowbrb s ≡n∑i=1αriαsiu2i +n∑i, j=1i< j(αriαsj + αrjαsi)uiu j−n∑i, j=1i< jαrjαsi[ui, u j] (mod A3).By assumption (i) of the statement we have that [ui, u j] ≡ 0 (mod A3), sobrb s ≡n∑i=1αriαsiu2i+n∑i, j=1i< j(αriαsj + αrjαsi)uiu j ≡ b sbr (mod A3). (5)Suppose brb s ∈ A3 for some r, s. Because of Eq. 5 and the assumptions (ii) and (iii)of the statement we have αriαsi = 0 and αriαsj + αsiαrj = 0 for every i, j. It follows thatαriαsj − αsiαrj = 0. Consequently, all of the order two minors formed by the k-th ands-th lines of the matrix M are zero, which is impossible because det M = 0. Hencebrb s, b sbr /∈ ω(L)3 and brb s ≡ b sbr (mod ω(L)3) for every r, s. By property(F-II) of the f.m. bases we conclude that brb s = b sbr. Thus A is a commutativealgebra, a contradiction. unionsqProof of the Theorem 2 Let S be a minimal set of generators of L as a restricted Liealgebra. Then, as L is powerful, by Lemma 1 we conclude that S is a minimal set ofthe nilpotent associative algebra ω(L) satisfying the hypotheses of Lemma 2, and theclaim follows. unionsqProof of the Theorem 3 Suppose, by contradiction, that u(L) has an f.m. basis B1with 1 ∈ B1, so that B = B1\{1} is an f.m. basis of ω(L) = rad(u(L)). Put n =dimF D1(L)/D2(L) and write B\ω(L)2 = {b 1, . . . , b n}. Consider an F-basis B ofFiltered Multiplicative Bases of Restricted Enveloping AlgebrasL as in the statement of Lemma 1 and let u1, . . . , un be the elements of B havingheight 1. Thus, by Lemma 1(ii), the set {u j + ω(L)2| j = 1, . . . , n} forms an F-basis ofω(L)/ω(L)2. Then, for every k = 1, . . . , n there exist αk1, . . . , αkn ∈ F such thatb k ≡n∑i=1αkiui (mod ω(L)2), (k = 1, . . . , n).Set u¯k = ∑ni=1 αkiui. Clearly, {u¯1, . . . , u¯n} is an F-linearly independent set whichgenerates L as a restricted subalgebra.Now, if L is powerful then, by Theorem 2, u(L) cannot have any f.m. basis, acontradiction. Therefore L′  L[p] and so there exist 1 ≤ r < s ≤ n such that theelement crs = [u¯r, u¯s] is not in L[p]. Since L is nilpotent of class 2, we have thatD2(L) = L′ + L[p] ⊃ L[p] = D3(L),hence crs has height two. Furthermore, one hasb 2s ≡ u¯2s (mod ω(L)3); b sbr ≡ u¯ru¯s − crs (mod ω(L)3).Since L is nilpotent of class 2, it follows thatbrb 2s ≡ u¯ru¯2s (mod ω(L)4);b surb s ≡ u¯ru¯2s − u¯scrs (mod ω(L)4);b 2s br ≡ u¯ru¯2s − [u¯r, u¯2s ] = u¯ru¯2s − 2u¯scrs (mod ω(L)4).Therefore the elementsv1 = brb 2s , v2 = b 2s br and v3 = b sbrb sare F-linearly dependent modulo ω(L)4. In view of property (F-I),(B ∩ ω(L)3)\ω(L)4is an F-basis for ω(L)3 modulo ω(L)4. Consequently, it follows that either vi ∈ ω(L)4for some i ∈ {1, 2, 3} or v j ≡ vk(mod ω(L)4) for some j, k ∈ {1, 2, 3}. In each case wehave a contradiction to Lemma 1, and the proof is complete. unionsqWe remark that the previous result fails without the assumption on the charac-teristic of the ground field. Indeed, let F be a field of characteristic 2 and considerthe restricted Lie algebra L = Fa + Fb + Fc with [a, b ] = c, [a, c] = [b , c] = 0, anda[2] = b [2] = c[2] = 0. Then it is straightforward to show that{1, a, b , ab , ab + c, ac, bc, abc}is an f.m. basis of u(L).Acknowledgement The authors are grateful to the referee for pointing out a problem in theoriginal proof of Theorem 1.V. Bovdi et al.References1. Bahturin, Y., Mikhalev, A.A., Petrogradsky, V.M., Zaicev, M.V.: Infinite-dimensional Lie super-algebras. In: de Gruyter Expositions in Mathematics, vol. 7. Walter de Gruyter, Berlin (1992)2. Balogh, Z.: On existing of filtered multiplicative bases in group algebras. Acta Math. Acad.Paedagog. Nyházi. (N.S.) 20(1), 11–30 (2004, electronic)3. Balogh, Z.: Further results on a filtered multiplicative basis of group algebras. Math. Commun.12(2), 229–238 (2007)4. Bautista, R., Gabriel, P., Roı˘ter, A.V., Salmerón, L.: Representation-finite algebras and multi-plicative bases. Invent. Math 81(2), 217–285 (1985)5. Bovdi, V.: On a filtered multiplicative basis of group algebras. Arch. Math. (Basel) 74(2), 81–88(2000)6. Bovdi, V.: On a filtered multiplicative bases of group algebras. II. Algebr. Represent. Theory6(3), 353–368 (2003)7. Bovdi, V.: Erratum to “On a filtered multiplicative basis...” Arch. Math. (Basel) 74(2), 81–88(2000, submitted)8. Feldvoss, J., Strade, H.: Restricted Lie algebras with bounded cohomology and related classes ofalgebras. Manuscr. Math. 74(1), 47–67 (1992)9. Gabriel, P., Roı˘ter, A.V.: Representations of finite-dimensional algebras. In: Algebra, VIII.Encyclopaedia Math. Sci., vol. 73, pp. 1–177. Springer, Berlin (1992, with a chapter by B. Keller)10. Kupisch, H.: Symmetrische algebren mit endlich vielen unzerlegbaren darstellungen. I. J. ReineAngew. Math. 219, 1–25 (1965)11. Landrock, P., Michler, G.: Block structure of the smallest Janko group. Math. Ann. 232(3), 205–238 (1978)12. Lincoln, M., Towers, D.: Frattini theory for restricted Lie algebras. Arch. Math. (Basel) 45(5),451–457 (1985)13. Riley, D.M., Shalev, A.: Restricted Lie algebras and their envelopes. Can. J. Math. 47(1), 146–164(1995)14. Roiter, A.V., Sergeichuk, V.V.: Existence of a multiplicative basis for a finitely spaced moduleover an aggregate. Ukr. Math. J. 46(5), 604–617 (1995, 1994)15. Siciliano, S., Weigel, T.: On powerful and p-central restricted Lie algebras. Bull. Aust. Math. Soc.75(1), 27–44 (2007)16. Strade, H., Farnsteiner, R.: Modular Lie algebras and their representations. In: Monographs andTextbooks in Pure and Applied Mathematics, vol. 116. Marcel Dekker, New York (1988)

Filtered multiplicative bases of restricted enveloping algebras

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