On Lie solvable restricted enveloping algebras

View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Algebra 314 (2007) 226–234 www.elsevier.com/locate/jalgebra On Lie solvable restricted enveloping algebras Salvatore Siciliano Dipartimento di Matematica “E. De Giorgi”, Università del Salento, Via Provinciale Lecce-Arnesano, 73100 Lecce, Italy Received 12 August 2006 Available online 7 April 2007 Communicated by Aner Shalev Abstract In this note we study the Lie derived lengths of a restricted enveloping algebra u(L), for a non-abelian restricted Lie algebra L over a field of positive characteristic p. For p > 2 we show that if the Lie derived length of u(L) is minimal then u(L) is Lie nilpotent. Moreover, we investigate the case when the strong Lie derived length of u(L) is minimal. For odd p we establish a classification of Lie centrally metabelian restricted enveloping algebras.  2007 Elsevier Inc. All rights reserved. Keywords: Restricted enveloping algebra; Lie derived length; Lie centrally metabelian 1. Introduction and statement of the main results Let R be a unital associative algebra over a field F . Recall that R can be viewed as a Lie algebra with Lie multiplication defined by [x, y] = xy − yx, for all x, y ∈ R. For subspaces A, B ⊆ R, we denote by [A, B] the linear span of all elements [a, b], with a ∈ A and b ∈ B. The Lie derived series of R is defined inductively by δ [0] (R) = R and δ [n+1] (R) = [δ [n] (R), δ [n] (R)]. Moreover, following [9], we consider the series of associative ideals of R defined by δ (0) (R) = R and δ (n+1) (R) = [δ (n) (R), δ (n) (R)]R. We say that R is Lie solvable (respectively strongly Lie solvable) if δ [n] (R) = 0 (δ (n) (R) = 0) for some n. In this case, the minimal n with such a property is called the Lie derived length (respectively strong Lie derived length) of R and denoted by dlLie (R) (dlLie (R)). Clearly, strong Lie solvability implies Lie solvability of R (and dlLie (R)  E-mail address: salvatore.siciliano@unile.it. 0021-8693/$ – see front matter  2007 Elsevier Inc. All rights reserved. doi:10.1016/j.jalgebra.2007.03.032 S. Siciliano / Journal of Algebra 314 (2007) 226–234 227 dlLie (R)), but the converse is in general not true. Moreover, R is called Lie nilpotent if it is nilpotent as a Lie algebra. Let L be a restricted Lie algebra over a field F of characteristic p > 0 and denote by u(L) the restricted (universal) enveloping algebra of L. An element x of L is p-nilpotent if there exists a m non-negative integer m such that x [p] = 0; a subset S of L is p-nilpotent if there is an m such m m that S [p] = {x [p] | x ∈ S} = 0. In the main theorem of [5], D. Riley and A. Shalev characterized the Lie solvable restricted enveloping algebras for odd p. On the other hand, in [13] the author proved (without restrictions on the characteristic) that u(L) is strongly Lie solvable if and only if the commutator subalgebra L′ of L is finite-dimensional and p-nilpotent. It turns out that for p > 2 the Lie solvability of u(L) is equivalent to the strongly Lie solvability (this is no longer true for p = 2: see Example 1 in [13]). Moreover, if u(L) is strongly Lie solvable then it is possible to have dlLie (u(L)) < dlLie (u(L)) (see Example 2 of [13]). While it is possible to compute the Lie nilpotency classes of u(L) by fairly satisfactory methods (see [6,15]), the computation of the Lie derived lengths represents a harder task. For more results on this topic the reader is referred to [7,13,14]. The purpose of this paper is to present some further contributions to this problem. In [13], the minimal value for the Lie derived length of non-commutative restricted enveloping algebras was determined. Indeed, it was proved that if L is not abelian then     dlLie u(L)  log2 (p + 1) and such lower bound is actually attained for every p. In Section 2 we investigate the structure of L when the Lie derived length or strong Lie derived length of u(L) coincides with this lower bound. Our first main result is the following. Theorem 1. Let L be a restricted Lie algebra over a field F of characteristic p > 2. If dlLie (u(L)) = ⌈log2 (p + 1)⌉ then u(L) is Lie nilpotent. It should be noted that Theorem 1 does not hold in characteristic 2 (cf. [14]). In the next theorem we characterize the non-commutative restricted enveloping algebras with minimal strong Lie derived length. Theorem 2. Let L be a restricted Lie algebra over a field F of characteristic p > 0. Then dlLie (u(L)) = ⌈log2 (p + 1)⌉ if and only if one of the following conditions is satisfied: (i) p = 2, dimF L′  2, L′ is central, and L′ [p] = 0; (ii) p = 2, dimF L′ = 1, and L′ [p] = 0; (iii) p > 2, dimF L′ = 1, L′ is central, and L′ [p] = 0. We recall that an associative algebra R is called Lie metabelian if δ [2] (R) = 0, and Lie centrally metabelian if [δ [2] (R), R] = 0. The Lie metabelian restricted enveloping algebras were characterized in [14]. Moreover, for p > 3 D. Riley and V. Tasić showed that u(L) is Lie centrally metabelian if and only if L is abelian (see [7]). In Section 3, we complete the characterization in odd characteristic by solving the (more difficult) case p = 3. Actually, we obtain the following result. 228 S. Siciliano / Journal of Algebra 314 (2007) 226–234 Theorem 3. Let L be a restricted Lie algebra over a field F of characteristic p > 2. Then u(L) is Lie centrally metabelian if and only if either L is abelian or all of the following conditions hold: p = 3, dimF L′ = 1, L′ is central, and L′ [p] = 0. As a consequence of such a result, for p > 2 a Lie centrally metabelian restricted enveloping algebra is in fact Lie metabelian. This is no longer true for p = 2. Finally, we mention that all the questions considered in the present paper arise also in the theory of group algebras and have been investigated by several authors (see [3,8–12,16]). It is interesting that the natural theoretic analogues of Theorems 1–3 do not hold for this class of algebras: we refer the reader to [3,11,12,16]. 2. Minimal Lie derived lengths The notations used throughout this paper are essentially standard. Let L be a restricted Lie algebra over a field F of positive characteristic p. We adopt the left-normed convention for longer commutators. For x, y ∈ L, we write [x,n y] to mean [x, y, . . . , y], where y appears in the latter expression n times. For a subset S of L we denote by Sp the restricted subalgebra generated by S. Note that, if I is an ideal then Ip is a restricted ideal of L. If an element x of L is n p-nilpotent, the minimal non-negative integer n such that x [p] = 0 is denoted by e(x). We write ω(L) for the augmentation ideal of u(L), namely, the associative ideal generated by L in u(L). It is well known that ω(L) is nilpotent if and only if L is finite-dimensional and p-nilpotent (see [5, Lemma 2.4]): in this case the minimal m such that ω(L)m = 0 is denoted by t (L). Proof of Theorem 1. Without loss of generality, we may assume F to be algebraically closed. In view of Theorem 1.3 of [5], L′ is finite-dimensional and p-nilpotent. Hence, by Theorem 1.1 of [5], it is enough to prove that L is nilpotent. Assume, if possible, that this is not true. Since dimF L′ < ∞, Lemma 2.3 of [4] implies that L/ζ2 (L) is finite-dimensional as well, where ζ2 (L) is the second term of the ascending central series of L. By Engel’s Theorem, it follows that L cannot satisfy any Engel condition. Consider a, b ∈ L such that [a,n b] = 0 for every n ∈ N and denote by H the Lie subalgebra generated by a and b. As dimF L′ < ∞, we have also dimF H < ∞. Since F is algebraically closed and H is not nilpotent, it follows that H contains a 2-dimensional non-abelian subalgebra. As a consequence we can choose x and y in H in such a way that [x, y] = x. By standard calculations (see relation (6) in the proof of Theorem 1 in [13]), for all non-negative integers r1 , r2 , s1 , s2 we have    r s r s (1) x 1 y 1 , x 2 y 2 = x r1 +r2 (y − r2 )s1 y s2 − (y − r1 )s2 y s1 . h We claim that, for all non-negative integers h and k satisfying k < p − h, the element x 2 y k is contained in δ [h+1] (u(L)). We proceed by induction on h. Suppose first h = 0. For every 0  k  p − 1, we have  k  xy , y = xy k and then xy k ∈ δ [1] (u(L)). Assume now h  1. The inductive hypothesis implies that δ [h] (u(L)) h−1 contains all elements x 2 y ν with 0  ν  p − h. Relationship (1) yields  2h−1 h−1  h y, x 2 = −2h−1 x 2 , x S. Siciliano / Journal of Algebra 314 (2007) 226–234 229 h so that x 2 ∈ δ [h+1] (u(L)), since p = 2. By (1) one has h   2h−1 2 2h−1  h y ,x = x 2 −2h y + 22(h−1) x h and thus, as x 2 ∈ δ [h+1] (u(L)) and p = 2, it follows that x 2 y ∈ δ [h+1] (u(L)). Suppose we have h h h already shown that δ [h+1] (u(L)) contains all elements x 2 , x 2 y, . . . , x 2 y k−1 . By (1), one has k+1   2h−1 k+1 2h−1  h  y ,x = x 2 y − 2h−1 − y k+1 x  k k + 1 (h−1)(k+1−j ) j 2h 2 y . =x (−1)k+1−j j j =0   2h k h Since x 2 y r ∈ δ [h+1] (u(L)) for every 0  r < k, also the element 2h−1 k+1 k x y is contained k+1 h [h+1] in δ (u(L)). Since p = 2 and p does not divide k = k + 1, we can conclude that x 2 y k ∈ δ [h+1] (u(L)), completing the inductive step. [n+1] (u(L)) = 0. Notice that 0 < p − 2n < p − n. Hence the Put n = ⌈log2 ( p+1 2 )⌉, so that δ n n previous part of the proof implies that the element x 2 is contained in δ [n+1] (u(L)). Thus x 2 = 0 and so, as 2n < p, this yields a contradiction to the PBW Theorem for restricted Lie algebras (see, e.g., Theorem 5.6 in Chapter 2 of [17]), and completes the proof. ✷ Remark 1. If L is a non-nilpotent restricted Lie algebra over a field F of characteristic p > 2, then the previous result implies that dlLie (u(L))  ⌈log2 2(p + 1)⌉. Actually, such a lower bound is the best possible. It is attained, for instance, if L is the non-abelian 2-dimensional restricted Lie algebra. More generally, if dimF L′ = 1 and L′ [p] = 0, then one has t (L′p ) = p. According to Lemma 1 of [13], we get (recall that p = 2):        dlLie u(L)  log2 2t L′p = log2 2(p + 1) . Thus dlLie (u(L)) = ⌈log2 2(p + 1)⌉. Proof of Theorem 2. Suppose that dlLie (u(L)) = ⌈log2 (p + 1)⌉. If p = 2, then, by [14], L satisfies one of the conditions (i) or (ii) of the statement. Assume that p > 2. An easy verification by induction (cf. [1, Proposition 3.8]) shows that, for every positive integer n, one has    2n −1 ω L′p ⊆ δ (n) u(L) . Hence     log2 t L′p + 1  dlLie u(L). (2) From Theorem 1.3 of [5] we already know that L′ is p-nilpotent. We claim that the p-map [p] acts trivially on L′ . Suppose, if possible, that this is not true. Then there exists x ∈ L′ such that e(x) > 1. By Theorem 3.4 of [6] and Lemma 1 of [15], one has     p 2  p e(x) = t {x}p  t L′p . 230 S. Siciliano / Journal of Algebra 314 (2007) 226–234 From elementary arithmetical considerations and from (2) it follows that      dlLie u(L)  log2 p 2 + 1 > log2 (p + 1) , a contradiction. Therefore L′ [p] = 0 and, in particular, L′ is a restricted subalgebra of L. Furthermore, as L′ is finite-dimensional (by Theorem 1.3 of [5]), by Engel’s Theorem L′ is nilpotent. Suppose that dimF L′ > 1. Thus, since L′ is nilpotent it contains a 2-dimensional abelian restricted subalgebra H . Hence t (L′ )  t (H ) = 2p − 1 and then, since p > 2, by (2) one has dlLie (u(L)) > ⌈log2 (p + 1)⌉, a contradiction. Finally, by Theorem 1, L is nilpotent and thus [L, L′ ] = 0, completing the first part of the proof. Conversely, if L verifies one of the conditions of the statement, the claim follows from [14] for even p, and by Proposition 3 of [13] for odd p. ✷ A unital associative algebra A is said to be strongly Lie nilpotent if A(m) = 0, where A(1) = A and A(m+1) = [A(m) , A]A. The smallest m such that A(m+1) = 0 is denoted by clLie (A). For arbitrary associative algebras, the strong Lie nilpotency is a stronger condition than Lie nilpotency (see [2]). Nevertheless, for restricted enveloping algebras these properties turn out to be equivalent (see [5]). Furthermore, if clLie (u(L)) denotes the ordinary nilpotency class of u(L) regarded as a Lie algebra, then clLie (u(L))  clLie (u(L)) and equality holds provided p > 3 (see [6]). As a consequence of Theorem 2 we shall see that, for odd p, the strong Lie derived length of a non-commutative restricted enveloping algebra is minimal if and only if its strong Lie nilpotency class is minimal, namely clLie (u(L)) = p (cf. [15]). In fact, the following holds Corollary 1. Let L be a non-abelian restricted Lie algebra over a field F of characteristic p > 2. Then dlLie (u(L)) = ⌈log2 (p + 1)⌉ if and only if clLie (u(L)) = p. Proof. Consider the chain of restricted ideals of L defined inductively by D(1) (L) = L, D(2) (L) = L′p ,     D(m+1) (L) = D(⌈ m+p ⌉) (L)[p] p + D(m) (L), L p (m  2). According to [6], if u(L) is strongly Lie nilpotent one has   clLie u(L) = 1 + (p − 1) md(m+1) (3) m1 where d(m) = dimF (D(m) (L)/D(m+1) (L)). Suppose, first, dlLie (u(L)) = ⌈log2 (p + 1)⌉. By Proposition 6.2 of [5] and Theorem 1, u(L) is strongly Lie nilpotent. Moreover, Theorem 2 implies that D(m) (L) = 0 for every m > 2. Thus, by (3), we conclude that clLie (u(L)) = p. Conversely, if clLie (u(L)) = p then by (3) we have necessarily d(2) = 1 and d(m) = 0 for every m > 2. This forces dimF L′p = 1, [L′p , L] = 0 and L′p [p] = 0. Hence Theorem 2 yields the claim. ✷ S. Siciliano / Journal of Algebra 314 (2007) 226–234 231 3. Lie centrally metabelian restricted enveloping algebras For an associative algebra A, we denote by ζ (A) the center of A and by γr (A) the terms of the Lie descending central series of A (defined inductively by γ1 (A) = A and γr+1 (A) = [γr (A), A]). In the proof of Theorem 3, we shall make use of the following result stated in [7]: Lemma 1. Let L be a restricted Lie algebra over a field of characteristic p > 2. If [[γn (u(L)), γr (u(L))], u(L)] = 0 for some n, r  1, then L is nilpotent. Proof of Theorem 3. In view of Theorem 2, the conditions expressed in the statement are clearly sufficient. Conversely, suppose that u(L) is Lie centrally metabelian and L is not abelian. By [7] it is enough to consider the case p = 3. Because of Lemma 1, L is nilpotent; therefore there exist two non-commuting elements a and b of L such that z = [a, b] is central in L. It is easy to see that a, b and z are F -linearly independent. We claim that z[p] = 0. In fact, we have a 2 z = [a, a 2 b] ∈ δ [1] (u(L)) and bz = [ba, b] ∈ δ [1] (u(L)), so that 2az3 = [a 2 z, bz] ∈ δ [2] (u(L)). Since u(L) is Lie centrally metabelian, it follows that 0 = [az3 , b] = z4 , hence the PBW Theorem forces z[p] = 0, as claimed. In order to complete the proof, it will suffice to show that L′ = F z. Suppose, if possible, that ′ L = F z; then consider three cases. Case 1: there exists c ∈ L such that [a, c] ∈ / F z. Clearly, the elements a, b and c are F -linearly independent. Put t = [a, c] and v = [b, c]. Since u(L) is Lie centrally metabelian, we have and     [b, v]z = [bz, v] = [ba, b], [b, c] ∈ ζ u(L)     [a, v]z = [az, v] = [a, ab], [b, c] ∈ ζ u(L) . Moreover, as [a, ab2 ] = 2abz, we have One concludes that   [a, v]bz + a[b, v]z = [abz, v] ∈ ζ u(L) . (4)   0 = a, [a, v]bz + a[b, v]z = [a, v]z2 and, analogously, [b, v]z2 = 0. By the PBW Theorem, it follows that [a, v] = αz and [b, v] = βz for some α, β ∈ F . As z ∈ ζ (u(L)), the Jacobi identity yields     [b, t] = b, [a, c] = a, [b, c] = [a, v] = αz. We claim that b and t commute. Assume the contrary, so that α = 0. Relation (4) yields   0 = [a, v]bz + a[b, v]z, c = (αv + βt)z2 . (5) 232 S. Siciliano / Journal of Algebra 314 (2007) 226–234 Therefore, the PBW Theorem forces αv + βt = kz for some k ∈ F . As α = 0, v is an F -linear combination of the elements t and z. In particular, v and t commute and then, by (5) and the fact that u(L) Lie centrally metabelian:     αzv = [b, t]v = [bv, t] = [b, bc], [a, c] ∈ ζ u(L) . As a consequence, 0 = [a, αzv] = αz[a, v] = α 2 z2 . Since α = 0, this latter relation violates the PBW Theorem; hence [b, t] = 0. If [a, t] = 0 one has        2at 2 z = a 2 z, ct = a, a 2 b , [ca, c] ∈ ζ u(L) and, as [b, t] = 0, this forces   0 = at 2 z, b = z2 t 2 . (6) Since t and z are F -linearly independent, relation (6) yields a contradiction to the PBW Theorem. On the other hand, if [a, t] = 0 notice that     [a, t]t = [at, t] = [a, ac], [a, c] ∈ ζ u(L) ; (7) thus     0 = [a, t]t, t = [a, t], t t. By the PBW Theorem, it follows that   [a, t], t = 0. (8) If [a, t] = λz for some λ ∈ F , then by (7) one has   0 = a, [a, t]t = λ2 z2 . Hence, the PBW Theorem forces λ = 0. Therefore [a, t] = 0, a contradiction. Therefore, [a, t] and z are F -linearly independent. One has     a[a, t]z = [az, at] = [a, ab], [a, ac] ∈ ζ u(L) . As a consequence, by (8) one obtains   0 = a[a, t]z, t = [a, t]2 z. By the linear independence of [a, t] and z, the last relation contradicts the PBW Theorem, and the proof for the first case is complete. Case 2: there exists c ∈ L such that [b, c] ∈ / F z. The proof is analogous to the previous case. Case 3: for every c ∈ L, [a, c] and [b, c] belong to F z. Let x, y ∈ L such that [x, y] and z are F -linearly independent. Put w = [x, y]. By assumption, the elements [a, x], [a, y], [b, x] and S. Siciliano / Journal of Algebra 314 (2007) 226–234 233 [b, y] are in F z. If [a, x] = µz for some µ ∈ F \{0}, an argument similar to the one used in case 1 (replacing b by x) yields a contradiction. Using a similar argument also for the other elements, we may assume that [a, x] = [a, y] = [b, x] = [b, y] = 0. As a consequence, the Jacobi identity yields and, analogously, [b, w] = 0. Finally, one concludes and hence   [a, w] = a, [x, y] = 0     azw 2 = [aw, abw] = [ax, y], [abx, y] ∈ ζ u(L)   0 = azw 2 , b = z2 w 2 . 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