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1 This article was downloaded by: [Universita degli Studi Roma Tre] On: 4 February 0, At: :56 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: Registered office: Mortimer House, 37-4 Mortimer Street, London WT 3JH, UK Communications in Algebra Publication details, including instructions for authors and subscription information: Deformations of the Restricted Melikian Lie Algebra Filippo Viviani a a Institute of Mathematics, Humboldt Universität of Berlin, Berlin, Germany Available online: 06 Jun 009 To cite this article: Filippo Viviani (009: Deformations of the Restricted Melikian Lie Algebra, Communications in Algebra, 37:6, To link to this article: PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae, and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand, or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material.

2 Communications in Algebra, 37: , 009 Copyright Taylor & Francis Group, LLC ISSN: print/53-45 online DOI: 0.080/ DEFORMATIONS OF THE RESTRICTED MELIKIAN LIE ALGEBRA Key Words: Filippo Viviani Institute of Mathematics, Humboldt Universität of Berlin, Berlin, Germany We compute the infinitesimal deformations of the restricted Melikian Lie algebra in characteristic 5. Cartan-type simple Lie algebras; Deformations; Restricted Lie algebras. 000 Mathematics Subject Classification: Primary 7B50; Secondary 7B0, 7B56.. INTRODUCTION The restricted Melikian Lie algebra M is a restricted simple Lie algebra of dimension 5 defined over the prime field of characteristic p = 5. It was introduced by [6], and it is the only exceptional simple Lie algebra which appears in the classification of restricted simple Lie algebras over a field of characteristic p 3 (see for p>7 and 7, for p = 5 7. The classification problem remains still open in characteristic and 3, where several exceptional simple Lie algebras are known (see 9, p. 09. This article is devoted to the study of the infinitesimal deformations of the restricted Melikian Lie algebra. The infinitesimal deformations have been computed for the other restricted simple Lie algebras in characteristic p 5. [8] proved that the simple Lie algebras of classical type are rigid, in analogy of what happens in characteristic zero. On the other hand, the author computed the infinitesimal deformations of the four families (Witt Jacobson, Special, Hamiltonian, Contact of restricted simple algebras of Cartan-type (see [0, ], showing that these are never rigid. By standard facts of deformation theory, the infinitesimal deformations of a Lie algebra are parametrized by the second cohomology of the Lie algebra with values in the adjoint representation. Assuming the notations from Section about the restricted Melikian algebra M as well as the definition of the squaring operator Sq, we can state the main result of this article. Received April 0, 007; Revised March 7, 008. Communicated by D. K. Nakano. Address correspondence to Filippo Viviani, via Marconi 36, Castorano 63030, Italy; viviani@math.hu-berlin.de 850

3 DEFORMATIONS OF THE RESTRICTED MELIKIAN LIE ALGEBRA 85 Theorem.. The infinitesimal deformations of the Melikian algebra M are given by H M M =Sq F SqD i F Sq D i F i= i= The strategy of our proof consists in exploiting the Hochschild Serre spectral sequence relative to the subalgebra of negative elements of M. A similar strategy has been used by [5] in order to prove that the Melikian algebra does not admit filtered deformations. As a byproduct of our proof, we give a new proof (see Theorem 3. of the vanishing of the first cohomology group of the adjoint representation (5, Proposition..3. The referee suggested another possible approach to prove the above Main Theorem. Namely, the restricted Melikian algebra M can be realized as a subalgebra of the restricted contact algebra K5 of rank 5 (see 6 in such a way that the negative elements of M coincide with the negative elements of K5. This allows to reduce the above result to the analogous result for the contact algebras (see. The article is organized as follows. In Section we recall, in order to fix the notations, the basic properties of the restricted Melikian algebra, the Hochschild Serre spectral sequence and the squaring operators. In Section 3 we prove that the cocycle appearing in the Main Theorem. are independent in H M M, and we outline the strategy to prove that they generate the whole second cohomology group. Each of the remaining four sections is devoted to carry over one of the four steps of this strategy.. NOTATIONS.. The Restricted Melikian Algebra M We fix a field F of characteristic p = 5. Let A = Fx x /x p xp be the F-algebra of truncated polynomials in variables, and let W = Der F A the restricted Witt Jacobson Lie algebra of rank. The Lie algebra W is a free A-module with basis D = and D x =. Let W x be a copy of W, and for an element D W we indicate with D the corresponding element inside W. The Melikian algebra M is defined as M = A W W with Lie bracket defined by the following rules (for all D E W and f g A: D Ẽ = D E + divdẽ D f = Df divdf f D + f D g D + g D = f g f g f Ẽ = fe f g = gd f fd g D + fd g gd f D

4 85 VIVIANI where divf D + f D = D f + D f A. The Melikian algebra M is a restricted simple Lie algebra of dimension 5 (see 9, Section 4.3 with a -grading given by (for all D E W and f A: deg M D = 3 degd deg M Ẽ = 3 dege + deg M f = 3 degf The lowest terms of the gradation are M 3 = FD + FD M = F M = F D + F D M 0 = ij= Fx i D j while the highest term is M 3 = x 4 x4 FD + FD. Moreover, M has a /3 -grading given by M = A M 0 = W M = W The Melikian algebra M has a root space decomposition with respect to a canonical Cartan subalgebra. Proposition.. (a T M =x D F x D F is a maximal torus of M (called the canonical maximal torus. (b The centralizer of T M inside M is the subalgebra C M = T M x x F x 4 x3 D F x 3 x4 D F which is hence a Cartan subalgebra of M (called the canonical Cartan subalgebra. (c Let M = Hom 5 i= x id i 5 5 where 5 is the prime field of F. There is a Cartan decomposition M = M M, where every summand M has dimension 5 over F. Explicitly, x a M a a x a D i M a i a i x a D i M a + i a + i In particular, if E M then deg E 3 + mod 5. Proof. See [9, Section 4.3]... Cohomology of Lie Algebras If is a Lie algebra over a field F and M is a -module, then the cohomology groups H M can be computed from the complex of n-dimensional

5 DEFORMATIONS OF THE RESTRICTED MELIKIAN LIE ALGEBRA 853 cochains C n M (n 0, that are alternating n-linear functions f n M, with differential dc n M C n+ M defined by df 0 n = n i i f 0 i n i=0 + p<q p+q f p q 0 p q n (. where the sign ˆ means that the argument below must be omitted. Given f C n Mand, we denote with f the restriction of f to, that is the element of C n M given by f 0 n = f 0 n With this notation, the above differential satisfies the following useful formula (for any and f C n M: d f = df (. df = f df (.3 where each C n M is a -module by means of the action f n = f n n f i n (.4 As usual we indicate with Z n M the subspace of n-cocycles and with B n M the subspace of n-coboundaries. Therefore, H n M= Z n M/B n M. A useful tool to compute cohomology of Lie algebras is the Hochschild Serre spectral sequence relative to a subalgebra. Given a subalgebra <, one can define a decreasing filtration F j C n M j=0n+ on the space of n-cochains: F j C n M= f C n M f n = 0if n+ j This gives rise to a spectral sequence converging to the cohomology H n M, whose first level is equal to (see 4: i= E pq = H q C p / M H p+q M (.5 In the case where is an ideal of (which we indicate as, the above spectral sequence becomes E pq = H p /H q M H p+q M (.6 Moreover, for the second page of the first spectral sequence (.5, we have the equality E p0 = H p M (.7

6 854 VIVIANI where H M are the relative cohomology groups defined (by from the subcomplex C p M C p M consisting of cochains orthogonal to, that is, cochains satisfying the two conditions: f = 0 (.8 df = 0 or equivalently f = 0 for every (.9 Note that in the case where, the equality (.7 is consistent with the second spectral sequence (.6 because in that case we have H p M= H p /M. Suppose that a torus T acts on both and M in a way that is compatible with the action of on M, which means that t g m = t g m + t g m for every t T, g and m M. Then the action of T can be extended to the space of n-cochains by t f n = t f n n f t i n It follows easily from the compatibility of the action of T and formula (.3, that the action of T on the cochains commutes with the differential d. Therefore, since the action of a torus is always completely reducible, we get a decomposition in eigenspaces i= H n M= H n M (.0 where = Hom F T F and H n M = f H n M t f = tf if t T. A particular case of this situation occurs when T and T acts on via the adjoint action and on M via restriction of the action of. It is clear that this action is compatible and moreover the above decomposition reduces to H n M= H n M 0 where 0 is the trivial homomorphism (in this situation, we say that the cohomology reduces to homogeneous cohomology. Indeed, if we consider an element f Z n M, then by applying formula (.3 with = t T, weget 0 = df t = t f df t = tf df t from which we see that the existence of a t T such that t 0 forces f to be a coboundary. Now suppose that and M are graded and that the action of respects these gradings, which means that d M e M d+e for all e d 0. Then the space of cochains can also be graded: a homogeneous cochain f of degree d is a cochain such that f e en M e i +d. With this definition, the differential becomes of degree 0, and therefore we get a degree decomposition H n M= d H n M d (.

7 DEFORMATIONS OF THE RESTRICTED MELIKIAN LIE ALGEBRA 855 Finally, if the action of T is compatible with the grading, in the sense that T acts via degree 0 operators both on and on M, then the above two decompositions (.0 and (. are compatible and give rise to the refined weight-degree decomposition H n M= H n M d (. d We will use frequently the above weight-degree decomposition with respect to the action of the maximal torus T M M 0 of the restricted Melikian algebra M (see Proposition...3. Squaring Operation There is a canonical way to produce -cocycles in Z over a field of characteristic p>0, namely, the squaring operation (see 3. Given a derivation Z (inner or not, one defines the squaring of to be Sqx y = p i= i x p i y i!p i! Z (.3 where i is the i-iteration of. In [3] it is shown that Sq H is an obstruction to integrability of the derivation, that is, to the possibility of finding an automorphism of extending the infinitesimal automorphism given by. 3. STRATEGY OF THE PROOF OF THE MAIN THEOREM First of all, we show that the five cocycles Sq D i i=, Sq, SqD i i= are independent in H M M. Observe that the first two cocycles have degree 5, the third has degree 0, and the last two have degree 5. Therefore, according to the decomposition (., it is enough to show that the first two are independent, the third is nonzero, and the last two are independent in H M M. To prove the independence of the first two cocycles, we observe that (for i j while for a cochain g C M M 0 5, we have that Sq D i x i D j x i D j = D i (3. dgx i D j x i D j = x i D j gx i D j x i D j gx i D j gx i D j x i D j = 0 (3. since gx i D j = 0 by degree reasons, gx i D j = 0 by degree and homogeneity reasons and x i D j x i D j = 0. The third cocyle is nonzero because { Sqxi x i x j D i = D i Sqx i x 3 j D j = D i (3.3

8 856 VIVIANI while for a cochain g C M M 0 0 we have that { dgxi x i xj D i = gx i xj dgx i xj 3D j = gx i xj (3.4 Finally, the independence of the last two cocycles follows from { SqDi x i D j x 4 i x jd j = D j SqD i x i D j x 3 i x j D j = 0 (3.5 together with the fact that for a cochain g C M M 0 5 we have that { ( dg xi D j xi 4x j D [ i = xi D j g ( xi 4x ] jd j dg ( x i D j xi 3x j D ([ j = g xi D j xi 3x j D ] ( (3.6 j = g x 4 i x j D j The remaining of this article is devoted to show that the above five cocycles generate the cohomology group H M M, as stated in the Main Theorem.. We outline here the strategy of the proof that will be carried over in the next sections. The proof is divided in five steps: Step I: Step II: We prove in Corollary 4.4 that H M M = H M M <0 M We prove in Proposition 5. that H M M <0 M H M 0 M 3 where M 0 acts on M 3 =D D F via projection onto M 0 /M = M 0 followed by the adjoint representation of M 0 onto M 3. Step III: We prove in Corollary?? that H M 0 M 3 where M acts trivially on M 3. Step IV: We prove in Proposition 7. that Sq D i F H M M 3 M 0 i= H M M 3 M 0 = SqD i F Sq F i= As a byproduct of our Main Theorem, we obtain a new proof of the following result (5, Proposition..3; see also 9, Chapter 7. Theorem 3. [5]. H M M = 0.

9 DEFORMATIONS OF THE RESTRICTED MELIKIAN LIE ALGEBRA 857 Proof. The spectral sequence (4., together with Proposition 4., gives that H M M = E 0 = H M M <0 M The Proposition 5. gives that H M M <0 M H M 0 M 3 Using the spectral sequence (6., together with Propositions 6. and 6., we get the vanishing of this last group. 4. STEP I: REDUCTION TO M <0 -RELATIVE COHOMOLOGY In this section, we carry over the first step outlined in Section 3. To this aim, we consider the homogeneous Hochschild Serre spectral sequence associated to the subalgebra M <0 <M (see Section.: ( E rs = H s M 0 <0 C r M/M <0 M 0 H r+s M M 0 = H r+s M M (4. We adopt the following notation: given elements E E n M <0 and G M, we denote with G E E n the cochain of C n M <0 M whose only nonzero values are for any permutation S n. Proposition 4.. G E E n E E n = sgng In the above spectral sequence (4., we have ( E 0 0 = ( E 0 = 0 0 Proof. Consider the Hochschild Serre spectral sequence associated to the ideal M 3 M : E rs = H r M /M 3 H s M 3 M H r+s M M (4. The lowest term M 3 =D D F acts on M = A W W via its natural action on A and via adjoint representation on W and W. Hence, according to [0, Proposition 3.4 and Corollary 3.5], we have that (for s = 0 M <0 if s = 0 H s x4 G M 3 M= D x4 G D if s = F G M <0 x4 x4 G D D if s = F G M <0 Moreover, M /M 3 = F acts on the above cohomology groups H s M 3 M via its adjoint action on M <0 = D D D D, that is, = 0 D i = 0 D i = D i

10 858 VIVIANI Hence, using that the above Hochschild Serre spectral sequence (4. is degenerate since M /M 3 = F has dimension, we deduce that M 3 M if s = 0 H s x4 G M M= D x4 G D F H F if s = G M 3 M H M M x4 x4 G D D F x4 H D x4 H D if s = F G M 3 M H M M Finally, consider the homogeneous Hochschild Serre spectral sequence associated to the ideal M M <0 : E rs 0 = H r M <0 /M H s M M 0 H r+s M <0 M 0 (4.3 From the explicit description of above, one can easily check that the only nonzero terms and nonzero maps of the above spectral sequence that can contribute to H s M <0 M 0 for s = are ( E 0 = 0 F ( E 0 0 = D D ( E 0 0 = i= x i 4 D i F ( E F 0 = i= x 4 i D i D D F where the maps are given by the differentials. Using the relation D D =, it is easy to see that all the above maps are isomorphisms, and hence the conclusion follows. In the next proposition, we need the following lemma. Lemma 4.. We have that H M <0 M= kh x 4 k D h D k F i j D j D i F D D D D F Proof. From the (nonhomogeneous Hochschild Serre spectral sequence associated to the ideal M M <0 : ( E rs = H r M <0 /M H s M M H r+s M <0 M we deduce the exact sequence 0 H M <0 /M M M H M <0 M H M M M <0/M H M <0 /M M M

11 DEFORMATIONS OF THE RESTRICTED MELIKIAN LIE ALGEBRA 859 Using the computation of H s M M for s = 0 from the Proposition 4., it is easily seen that H ( M <0 /M M M ij D j D F = i ad F H M M M x <0/M = hk h 4D k D h F H M <0 /M M M = D D F F Using the relation D D =, it is easy to see that =, which gives the D D conclusion. Proposition 4.3. Proof. In the above spectral sequence (4., we have that E 0 = 0 We have to show the injectivity of the level differential map d ( E 0 ( E In the course of this proof, we adopt the following convention: given an element f C M <0 C s M/M <0 M, we write its value on D M <0 as f D C s M/M <0 M. We want to show, by induction on the degree of E M/M <0, that if df = 0 H M <0 C M/M <0 M, then we can choose a representative f of f H M <0 C M/M <0 M such that f D E = 0 for every D M <0. So suppose that we have already found a representative f such that f D F = 0 for every F M/M <0 of degree less than d and for every D M <0. First of all, we can find a representative f of f such that f Di E = j i x 4 i D j for i = j= f E = 0 f D E = D + D f D E = D D 0 ( for every E M of degree d. Indeed, by the induction hypothesis, the cocycle condition for f is f DD E = D f D E D f D E f DD E for any D D M <0. On the other hand, by choosing an element h C M/M <0 M that vanishes on the elements of degree less than d, we can add to f (without changing its cohomological class neither affecting the inductive assumption the coboundary h whose value on E are h D E = D he. Hence, for a fixed element E of degree d, the map D f D E gives rise to an element of H M <0 M and, by Lemma 4., we can chose an element he as above such that the new cochain f = f + h verifies the condition of above.

12 860 VIVIANI Note that, by the homogeneity of f, we have the following pairwise disjoint possibilities for E: j i 0 for i = E { x j xj+ x4 j x j+d j+ x3 j xj+ 3 D j x j xj+ 4 D j+} j 0 E { x j xj+ 4 x j+ D jxj 4x3 j+ D j+ x j 3 D j x } j x j+ D j+ 0 E { x 4x3 D x 3x4 D x x D x x D } Now we are going to use the condition that d f = 0 E 0, that is d f = g for some g C M/M <0 M 0. Explicitly, for A B M/M <0, we have that (for any D M <0 g D A B = D ga B gd A B ga D B (4.4 d f D A B = f D A B A f D B + B f D A f DA B + f DB A (4.5 where the last two terms in the first formula can be nonzero only if D A M 0 and D B M 0, respectively, where the last two terms in the second formula can be nonzero only if D A M <0 and D B M <0, respectively. Suppose first that j i 0 for a fixed j and for i =. Then it is straightforward to check that for each E in the above list it is possible to find two elements A B M such that A B = E, degb = 0, and A does not belong to the above list. Apply the above formulas for each such pair A B. Taking into account the inductive hypothesis on the degree and the homogeneity assumptions, the formula (4.5 becomes while the formula (4.4 gives d f Di A B = f Di A B = f Di E = j i x 4 i D j g Di A B = D i ga B gd i A B The first term in the last expression is a derivation with respect to D i, and therefore it cannot involve the monomial xi 4D j. The same is true for the second term as it follows by applying the formulas (4.5 and (4.4 for the elements add i k A (with k = p and B, and using the vanishing hypothesis: { d f Di add i k A B = 0 g Di add i k A B = D i gadd i k A B gadd i k+ A B Therefore, we conclude that j i = 0, an absurd. The other cases j 0 and 0 are excluded using a similar argument. Finally, we get the main result of this section. Corollary 4.4. We have that H M M = H M M <0 M

13 DEFORMATIONS OF THE RESTRICTED MELIKIAN LIE ALGEBRA 86 Proof. From the spectral sequence (4., using the vanishing of E 0 0 (Proposition 4. and of E 0 (Proposition 4.3, we get that H M M = ( E 0 0 = ( E 0 = H M M 0 <0 M 5. STEP II: REDUCTION TO M 0 -COHOMOLOGY In this section, we carry over the second step of proof of the main theorem (see Section 3. Consider the action of M 0 on M 3 =D D F obtained via projection onto M 0 /M = M 0 followed by the adjoint representation of M 0 onto M 3. Proposition 5.. Proof. We have that { H M M <0 M H M 0 M 3 H M M <0 M H M 0 M 3 For every s 0, consider the map s C s M M <0 M C s M 0 M 3 induced by the restriction to the subalgebra M 0 M and by the projection M M/M = M 3. It is straightforward to check that the maps s commute with the differentials, and hence they define a map of complexes. Moreover, the orthogonality conditions with respect to the subalgebra M <0 give the injectivity of the maps s. Indeed, on one hand, the condition (.8 says that an element f C s M M <0 M is determined by its restriction to s M 0. On the other hand, the condition (.9 implies that the values of f on an s-tuple are determined, up to elements of M M <0 = M 3, by induction on the total degree of the s-tuple. The map 0 is an isomorphism since C 0 M M <0 M= M M <0 = M 3 = C 0 M 0 M 3 From this, we get the first statement of the proposition. Moreover, it is easily checked that if g C M 0 M 3 is such that dg C M M <0 M, then g C M M <0 M. This gives the second statement of the proposition. 6. STEP III: REDUCTION TO M 0 -INVARIANT COHOMOLOGY In this section, we carry over the third step of the proof of the Main Theorem (see Section 3. To this aim, we consider the Hochschild Serre spectral sequence relative to the ideal M M 0 : E rs = H r M 0 H s M M 3 H r+s M 0 M 3 (6. The first line E 0 of the above spectral sequence vanish.

14 86 VIVIANI Proposition 6.. In the above spectral sequence (6., we have for every r 0 E r0 = H r M 0 M 3 = 0 Proof. Observe that, since the canonical maximal torus T M is contained in M 0, we can restrict to homogeneous cohomology (see Section.. But there are no homogeneous cochains in C r M 0 M 3. Indeed the weights that occur in M 3 are i while the weights that occur in M 0 are 0 and i j. Therefore the weights that occur in M k 0 have degree congruent to 0 modulo 5, and hence they cannot be equal to i. Next, we determine the groups E r = H r M 0 H M M 3 for r = of the above spectral sequence (6.. Proposition 6.. In the above spectral sequence (6., we have that E r = { 0 if r = 0 Sq D Sq D F if r = where Sq D i (for i = is the restriction of Sq D i to M 0 M. Proof. Using the Lemma 6.3 below, we have that H M M 3 = { f C M + M M 3 f x D = f x D } Observe that, since the canonical maximal torus T M is contained in M 0, we can restrict to homogeneous cohomology (see Section.. The vanishing of E 0 follows directly from the fact that there are no homogeneous cochains in C M 0 H M M 3. The elements Sq D i for i = belong to H M 0 H M M 3 and are non-zero in virtue of the formulas (3. and (3.. Moreover, it is easy to see that, for homogeneity reasons, C M 0 H M M 3 0 is generated by Sq D i (i =, which gives the conclusion. Lemma 6.3. M M = M 3 + x D + x D F Proof. Clearly, M M M and M M M = M M. Since M = x x F, we have that M M =x x F = x D + x D F. Hence the proof is complete if we show that M 3 M M. We will consider the /3grading on M, and we will consider separately M 3 M i, with i = 0. (i M 3 M 0 M M because of the formula x j x a D i = x j x a D i (ii M 3 M = M 4 M M M Indeed, the formula x a D i x i = a i x a shows that x a (with a belongs to M M provided that x a x 3x3, and this exceptional case is handled with the formula x x4 D x = 4x 3x3.

15 DEFORMATIONS OF THE RESTRICTED MELIKIAN LIE ALGEBRA 863 (iii M 3 M = M 5 M M M Indeed, the formula xi D i x a D j = x i D i x a D j + 4x i x a D j = a i + 4 ij x a+ i Dj shows the inclusion for the elements of M 5 M with the exception of the two elements of the form x i xj 4 D j for i j, for which we use the formula xi D j xi xj 4 D i = 4x 3 i xj 3 D i x i xj 4 D j. Finally, we get the result we were interested in. Corollary 6.4. We have that H M 0 M 3 Sq D i F H 3 M M 3 M 0 i= Proof. From the above spectral sequence (6., using the Propositions 6. and 6., we get the exact sequence 0 Sq D Sq D F H M 0 M 3 H M M 3 M 0 which gives the conclusion. 7. STEP IV: COMPUTATION OF M 0 -INVARIANT COHOMOLOGY In this section, we carry over the fourth and last step of the proof of the Main Theorem by proving the following proposition. Proposition 7.. We have that H M M 3 M 0 = SqD i F Sq F i= Proof. The strategy of the proof is to compute, step by step as d increases, the truncated invariant cohomology groups ( M0 H M M M 3 d+ Observe that if d 3, then M d+ = 0, and hence we get the cohomology we are interested in. The Lie algebra M has a decreasing filtration M d d=3, and the adjoint action of M 0 respects this filtration. We consider one step of this filtration M d = M d M d+ M M d+ and the related Hochschild Serre spectral sequence ( ( E rs = H r M H s M M d M 3 H r+s M M d M 3 (7. d+

16 864 VIVIANI We fix a certain degree d, and we study, via the above spectral sequence, how the truncated cohomology groups change if we pass from d to d +. It is easily checked that, by homogeneity, we have H ( M M0 M M 3 3 C M M 3 0 = 0 Therefore, for the rest of this section, we suppose that d 3. Observe also that, since M d is in the center of M /M d+, and M 3 is a trivial module, then H s( M d M 3 = C s M d M 3, and M /M d acts trivially on it. Consider the following exact sequence deduced from the spectral sequence (7. E 0 ( ( = H M M M 3 H M M d M 3 E 0 d+ From the above Lemma 6.3 (using that d 3, we get that the first two terms are equal to ( ( ( H M M M 3 = H M M M d M 3 = C M d+ M 3 + x D + x 3 D F and therefore we deduce that E 0 = 0. Together with the vanishing E0 in the Lemma 7. below, we deduce the following exact diagram: = 0 proved We take the cohomology with respect to M 0 and use the Lemmas Observe that the cocycle inv E M 0 which appears for d = 5 is annihilated by the differential of inv E 0 M 0 = C M d M 3 M 0 for d = 7. Moreover, the element x i D j ij inv H M 0 C M 7 M 3 does not belong to the kernel of the differential map d H M 0 C M 7 M 3 = H ( M 0 E 0 ( H M 0 E 0 Indeed the element x i D j ij inv does not vanish on T M, and the same is true for its image through the map d, while any coboundary of H M 0 E 0 must vanish on T M by homogeneity. Therefore, the only cocycles that contribute to the required cohomology group are Sq SqD SqD. The remaining part of this section is devoted to prove the lemmas that were used in the proof of the above proposition. In the first lemma, we show the vanishing of the term E 0 of the above spectral sequence (7..

17 DEFORMATIONS OF THE RESTRICTED MELIKIAN LIE ALGEBRA 865 Lemma 7.. In the above spectral sequence (7., we have E 0 3 = 0. Proof. By definition, E 0 3 is the kernel of the map ( d C M d M 3 = E 0 E = H M C M M d M 3 d that sends a -cochain f to the element df given by df EF G = fe F G whenever dege + degf = d, and 0 otherwise. The subspace of coboundaries B ( M M d C M d M 3 is the image of the map ( ( C M C M M d M 3 C M C M d M d M 3 d that sends the element g to the element g given by g EF G = g EF G Hence g vanishes on the pairs E F for which dege + degf = d. Therefore, if an element f C M d M is in the kernel of d, that is, df = g for some g as before, then it should satisfy fe F G = 0 for every E F G such that degg = d and dege + degf = d. By letting E vary in M and F in M d, the bracket E F varies in all M d by Lemma 6.3 (note that we are assuming d 3. Hence the preceding condition implies that f = 0. In the new two lemmas, we compute the M 0 -invariants and the first M 0 -cohomology group of the term E 0 = C M d M 3 in the above spectral sequence (7.. Lemma 7.3. Define inv M 7 M 3 by (i j: inv ( x 3 i x3 j D i = idi and inv ( x 4 i x j D i = idj where i = or if i = or i =, respectively. Then { inv if d = 7 C M d M 3 M 0 = 0 otherwise. Proof. By homogeneity, we can assume that d mod 5. We will consider the various cases separately. d=7 Since M 7 = A 3 =x 3x x x x x3 F with weights 3, 0 4, 4 0, 3, respectively, a homogeneous cochain g C M d M 3 M 0 can take the nonzero values gxi x j = a i D j for i j. We obtain the vanishing from the following M 0 -invariance condition: 0 = x j D i gx 3 i = g3x i x j = 3a i D j

18 866 VIVIANI d= A homogeneous cochain g C M M 3 0 can take the nonzero values: gxi 4x jd j = i j D i for i j. We get the vanishing of g by mean of the following cocycle condition: ( 0 = dg xi D j x 3 i x j D ( j = g x 4 i x jd j = i j D i (7. d=7 M 7 = W 5 = i jk x4 i x D j k F i jk x3 i x3 D j k with weights i j k and k, respectively. A homogeneous cochain g C M d M 3 M 0 can take the nonzero values: gxi 3x3 D j i = i D i and gxi 4x D j i = i D j for i j. The only M 0 -invariance conditions are the following: x i D j g ( xi x4 D ( j i = g 4x 3 i x 3 D j i xi x4 D j j = 4i + j D i x i D j g ( xi x4 D [ j j = xi D j g ( xi x4 D ] ( j j g 4x 3 i xj 3 D j = j 4 j D j x i D j g ( xi 3x3 D j i = i 3 i + j D j from which it follows that i = i = j = j, that is, g is a multiple of inv. d= Since M = A 8 =x 4x4 with weight, there are no homogeneous cochains. Lemma 7.4. We have that SqD i F if d = H M 0 C i= M d M 3 = x i D j ij inv F if d = 7 0 otherwise, where SqD i is the restriction of SqD i to M 0 M and inv M 7 M 3 is the cochain defined in Lemma 7.3. Proof. By homogeneity, we can assume that d mod 5 and consider the various cases separately. d=7 Consider a homogeneous cochain f C M 0 C M 7 M 3 0. By applying cocycle conditions of the form 0 = df xk D k x i D j, it follows that f xk D k C M 7 M 3 M 0 = 0 (by the preceding lemma. By homogeneity, f can take only the following nonzero values for i j: f xi D j xj 3 = b id i and f xi D j x i xj = c id j. By possibly modifying f with a coboundary dg (see Lemma 7.3, we can suppose that b i = b j = 0. The following cocycle condition ( 0 = df xi D j x j D i x i x [ ( ] ( ( j = xi D j f xj D xj i x i fxj D i x 3 i + fxi D xi j x j = c j b j + c i D j = c j + c i D j gives that c j = c i = 4c j, and hence that c i = c j = 0, that is, f = 0. d= First of all, the cocycles SqD i for i = belong to H M 0 C M M 3 0 since they are restriction of global cocycles, and, moreover, they are

19 DEFORMATIONS OF THE RESTRICTED MELIKIAN LIE ALGEBRA 867 independent as it follows from the formulas (3.5 and (3.6. It remains to show that the above cohomology group has dimension less than or equal to. Consider a homogeneous cocycle f C M 0 C M M 3 0. First of all, we observe that f must satisfy f xi D i = 0 Indeed, by the formula (.3, we have 0 = df xi D i = x i D i f df xi D i from which, since the first term vanishes for homogeneity reasons, it follows that f xi D i C M p M M 0 which is zero by the previous Lemma 7.3. Therefore, f can take only the following nonzero values (for i j: ( f xi D j x 3 i xj D j = ij D i ( f xi D j x 4 i x j D i = ij D i ( f xi D j x 4 i x j D j = ij D j By possibly modifying f with a coboundary (see formula (7., we can assume that ij = 0. The coefficients ij are determined by the coefficients ij in virtue of the following cocycle condition: ( 0 = df xi D j x j D i x 4 i x ( jd j = fxi D j x 3 i x j D ( j + fxi D j x 4 i x jd i + [ ( x j D i f xi D j x 4 i x ] jd j = ij ij + ij D i = ij + ij D i Therefore, the cohomology group depends on the two parameters and, and hence has dimension less than or equal to. d=7 Consider a homogeneous cochain f C M 0 C M 7 M 3 0. By imposing cocycle conditions of the form 0 = df xk D k x i D j, one obtains that f xk D k C M 7 M 3 M 0 =inv (by the preceding lemma. Put fxk D k = k inv. By homogeneity, the other nonzero values of f are (for i j: f xi D j ( x i x4 j D i = i D i f xi D j ( x i x4 j D j = i D j f xi D j ( x 3 i x3 j D i = i D j By possibly modifying f with a coboundary dg, we can assume that f x D = 0 (see Lemma 7.3. Considering all the cocycle conditions of the form 0 = df x D x D = x D f x D f x D + f x D, one gets 0 = df x D x D ( x 3 x D 3 = = df x D x D ( x 3 x D 3 = = df x D x D ( x x D 4 = + 0 = df x D x D ( x 4 x D = + Since the 4 4 matrix associated to the preceding system of 4 equations in the 4 variables,,, is invertible (it has determinant equal to, we conclude that f x D = 0 and f x D = f x D is a multiple of inv. d= There are no homogeneous cochains.

20 868 VIVIANI In the last lemma, we compute the M 0 -invariants of the term E of the above spectral sequence (7.. In view of Lemma 6.3 and the hypothesis d 3, we have that ( E = C M M x D + x D M dm 3 Lemma 7.5. In the above spectral sequence (7., we have that E M 0 = Sq if d = 6 F inv F if d = 5 0 otherwise, where Sq is the restriction of Sq to M M 6, and inv is defined by inv E F = inve F where inv M 7 M 3 is the M 0 -invariant map defined in Lemma 7.3. Proof. The term E M 0 = E 3 M 0 is the kernel of the map d ( E M0 ( E 30 ( M0 E 30 = H 3 M M M 3 d In order to avoid confusion, during this proof, we denote with f B 3( M M M d 3 (instead of the usual df the coboundary of an element f C ( M M M d 3. Note that M =x x F with weights, respectively, 4 3 and 3 4, while M /x D + x D =x D = x D x D x D F with weights, respectively,, 3, and 3. Note also that after the identification x D = x D, the action of M 0 = W 0 on M /x D + x D is given by (for i j x i D j x i D j = 0, x i D j x D = jx i D j and x i D j x j D i = jx D, where j = if, respectively, j =. By homogeneity, an M 0 -invariant cochain of E can assume nonzero values only on M M d if d mod5oronm M d if d 0 mod 5. We will consider the various cases separately. d=5 M 5 = W = ik x D i k F k x x D k F with weights + i k and 3 3 k, respectively. A homogeneous cochain g E M 0 can take only the following nonzero values (for i j: { ( g x D x i x j D i = hi D i g ( x D xj D j = ki D i g ( x i D j xj D i = li D i g ( x i D j x i x j D i = mi D j g ( x i D j xj D j = ni D j

21 DEFORMATIONS OF THE RESTRICTED MELIKIAN LIE ALGEBRA 869 Consider the following M 0 -invariance conditions: 0 = x i D j g ( x j D i x D j j = jki m j D i 0 = x i D j g ( x j D i x i x j D i = jhi n j + m j D i 0 = x i D j g ( x i D j x D j i = li m i + n i D j 0 = x i D j g ( x j D i x i x j D j = mj + jh j l j D j 0 = x i D j g ( x j D i xi D i = nj + jk j + l j D j From the first 3 equations, one obtains that m j n j l j = jk i h i + k i h i k i, and substituting in the last two equations, one finds that h i = h j = h and k i = k j = k. Suppose now that g is in the kernel of the map d, that is, dg = f for some f C M /M 5 M 3. Applying 0 = dg + f to the triples x i x j D j x i x j and x j x i D j x i x j for i j, we get the two conditions { ( ( f xi x j D j x i x j = ig xj D j xi D i = kdj f ( ( x i x j D j x i x j = jg xi D j xj D j = h + kdj from which it follows that h = 0. Considering now the triples x i x r D r x i x j and x i x r D r xj (for i j and some r =,weget { ( ( f xi x r D r x i x j = ig xr D r xi D i = irkdj f ( ( x i x r D r xj = ig xr D r x i x j D i xj D j = irkdi Finally, considering the triple x i x i D ix i x j D i, and using the two preceding relations, we get 0 = dg + fx i x i D ix i x j D i = fx i D ix i x i x j D i fx i x j D i x i x i D i = 3fx j x i D i x i fx i D ix i x j = 6kD j kd j = kd j from which it follows that k = 0, that is g = 0. d=6 First of all, the cocycle Sq is an element of E M 0 since it is the restriction of a global cocycle and is nonzero in virtue of formulas (3.3 and (3.4. Therefore, it remains to show that the dimension of E M 0 is at most. Consider a cochain g E M 0, which is in particular homogeneous. Since M6 = W = ik x i D k F k x x D k F with weights i k and k, respectively, g can take only the following nonzero values (for i j: { g ( x i x i x j D i = ci D i g ( x i x i x jd i = ei D j g ( x i x 3 j D j = di D i g ( x i x i x j D j = fi D j

22 870 VIVIANI Consider the following M 0 -invariance conditions: 0 = x i D j g ( x i xj 3 D ( i = g xi 3x i xj D i xj 3D j = 3ci + d i D i 0 = x i D j g ( x i x i xj D i = ci e i + f i D j 0 = x i D j g ( x i xj 3 D j = xi D j d i D i gx i 3x i xj D j = d i 3f i D i 0 = x i D j g ( x j x i xj D i = ci f j + e j D i From the first 3 equations, one gets f i c i d i = e i e i e i, and substituting into the last equation, one finds e i = e j = e. Therefore, E M 0 depends on one parameter and hence has dimension at most. d=0 M 0 = A 4 = i x4 i F i j x3 i x j F i j x i x j F with weights i j, i j and 0 0, respectively. Consider a cochain g E M 0. Since M 0 acts transitively on M, to prove the vanishing of g it is enough to prove that gx D = 0 By homogeneity the only possible nonzero such values are gx D x 3x and gx D x 4, and the vanishing follows from the M 0-invariance conditions: { 0 = x D g ( ( x D x x = g x D x 3x 0 = ( x D g ( x D x 3x [ = x D gx D x 3x ] g ( x D x 4 d=5 M 5 = W 5 = i jk x4 i x j D k F i jk x3 i x3 j D k F with weights i + j k and 3 i + 3 j k, respectively. A homogeneous cochain g E M 0 can take only the following nonzero values (for i j: { ( g x D xi 3x3 j D i = pi D i g ( x D xi x4 j D j = qi D i g ( x i D j xi x4 j D i = ri D i g ( x i D j xi 3x3 j D i = si D j g ( x i D j xi x4 j D j = ti D j Consider the following M 0 -invariance conditions: 0 = x i D j g ( x j D i xi x4 j D j = jqi s j D i 0 = x i D j g ( x j D i xi 3x3 j D i = jpi 3t j + s j D i 0 = x i D j g ( x i D j xi x4 j D i = ri + s i + t i D j 0 = x i D j g ( x j D i xi 3 x3 j D j = sj + jp j 3r j D j 0 = x i D j g ( x j D i xi 4x j D i = tj + jq j + r j D j From the first 3 equations, one gets that s j t j r j = j q i p i + q i p i q i, and substituting in the last two equations, one obtains p i = p j = p and q i = q j = q. Suppose now that g is in the kernel of d, that is, dg = f for some f C M /M 5 M 3. Applying 0 = dg + f to the triple x i x j D j x 4 i x j D i for i j, we get 0 = f ( x i x j D i x 4 i x j D i + g ( xi x 4 i x j D i x j D i = f ( xi x j D i x 4 i x j D i

23 DEFORMATIONS OF THE RESTRICTED MELIKIAN LIE ALGEBRA 87 Considering the triple x j x i D i xi 4x D j i, and using the preceding vanishing, we obtain 0 = f ( x i x j D i x 4 i x j D i g ( xi D i x 4 i x j D i = iqdj that is, q = 0. For p =, one obtains that g = inv, which is clearly in the kernel of d since it is the restriction of a cocycle on M M. d=6 M 6 = A 6 = i j x i x4 j F x 3x3 F with weights j and, respectively. A homogeneous cochain g E M 0 can take the nonzero values (for i j: gx i xi x4 j = a id i and gx i x 3x3 = b id j for i j. We get that a i = a j = b i = b j = a by considering the following M 0 -invariant conditions: { 0 = xi D j g ( x i xi x4 j = xi D j a i D i g ( x i 4xi 3x3 j = ai + b i D j 0 = x i D j g ( ( ( x j xi x4 j = g xi xi x4 j g xj 4xi 3x3 j = ai + b j D i Now suppose that g is in the kernel of the map d, that is, dg = f for some f C M /M 6 M 3. Applying the relation 0 = dg + f to the two triples x i x j xi 4x j D i and x i x j xi 3x3 j D i for i j, weget { ( f x D + x D xi 4x j D ( i = g xi xi 4x j D ( i x j g xj xi 4x j D i x i = a f ( x D + x D xi 3x3 j D ( i = g xi xi 3x3 j D ( i x j g xj xi 3x3 j D ( i x i = a Considering the triples x x D + x D x 3x D and x x D + x D x x3 D, and using, weget { ( ( f x D + x x D x 3x D = f x D + x D x 4x D ( g x x 3x3 = a f ( x D ( + x x D x x3 D = f x D + x D x 3x3 D ( + g x x 3x3 = 0 ( Finally, using the relations, we get the vanishing of g by mean of the following: 0 = dg + f ( x x D ([ + x x D x x3 = f x x D ] + x x D x x3 + f ([ ] x x x3 x D ([ + x x D g x D ] + x x D x x3 x = f ( x 3 x D + x x3 D x D ( + x x D g x 3 x3 x = a a = 3a d=0 M 0 = W 6 = i jk x4 i x3 D j k F with weights i k. A homogeneous cochain g E M 0 can take only the nonzero values gx i D j xi 4x3 D j j = i D i. The vanishing follows from the M 0 -invariance conditions 0 = x i D j g ( x i D j x 3 i x4 j D j = 4g ( xi D j x 4 i x3 j D j = i D i d= Since M = W 7 =x 4x4 D x 4x4 D F with weights 3 4 and 4 3, respectively, there are no homogeneous cochains.

24 87 VIVIANI ACKNOWLEDGMENTS During the preparation of this article, the author was supported by a grant from the Mittag Leffler Institute of Stockholm. The result presented here constitutes part of my doctoral thesis. I thank my advisor prof. Schoof for useful advices and constant encouragement. I thank the referee for useful suggestions that helped to improve and clarify the exposition. REFERENCES [] Block, R. E., Wilson, R. L. (988. Classification of the restricted simple Lie algebras. J. Algebra 4:5 59. [] Chevalley, C., Eilenberg, S. (948. Cohomology theory of Lie groups and Lie algebras. Trans. Amer. Math. Soc. 63:85 4. [3] Gerstenhaber, M. (964. On the deformation of rings and algebras. Ann. Math. 79: [4] Hochschild, G., Serre, J. P. (953. Cohomology of Lie algebras. Ann. Math. 57: [5] Kuznetsov, M. I. (990. On Lie algebras of contact type. Commun. Algebra 8: [6] Melikian, G. M. (980. Simple Lie algebras of characteristic 5 (Russian. Uspekhi Mat. Nauk 35: [7] Premet, A., Strade, H. (00. Simple Lie algebras of small characteristic. III: The toral rank case. J. Algebra 4: [8] Rudakov, A. N. (97. Deformations of simple Lie algebras (Russian. Izv. Akad. Nauk SSSR Ser. Mat. 35:3 9. [9] Strade, H. (004. Simple Lie algebras over fields of positive characteristic I: Structure theory. De Gruyter Expositions in Mathematics. Vol. 38. Berlin: Walter de Gruyter. [0] Viviani, F. (008. Infinitesimal deformations of restricted simple Lie algebras I. J. Alg. 30: [] Viviani, F. (009. Infinitesimal deformations of restricted simple Lie algebras II. J. Pure App. Alg. 3:70 7.