aa r X i v : . [ m a t h . A T ] J a n ON LEIBNIZ COHOMOLOGY
J ¨ORG FELDVOSS AND FRIEDRICH WAGEMANN
Abstract.
In this paper we prove the Leibniz analogue of Whitehead’s van-ishing theorem for the Chevalley-Eilenberg cohomology of Lie algebras. Asa consequence, we obtain the second Whitehead lemma for Leibniz algebras.Moreover, we compute the cohomology of several Leibniz algebras with ad-joint or irreducible coefficients. Our main tool is a Leibniz analogue of theHochschild-Serre spectral sequence, which is an extension of the dual of a spec-tral sequence of Pirashvili for Leibniz homology from symmetric bimodules toarbitrary bimodules.
Introduction
In [1], the authors study the cohomology of semi-simple Leibniz algebras, i.e.,the cohomology of finite-dimensional Leibniz algebras L with an ideal of squaresLeib( L ) such that the corresponding canonical Lie algebra L Lie := L / Leib( L ) issemi-simple, and conjecture that HL ( L , L ad ) = 0. In [16], the authors determinethe deviation of the second Leibniz cohomology of a complex Lie algebra with ad-joint or trivial coefficients from the corresponding Chevalley-Eilenberg cohomology.With these motivations in mind, we systematically transpose Pirashvili’s resultsand tools from homology (see [31]) to cohomology, generalize the dual of one ofPirashvili’s spectral sequences from symmetric bimodules to arbitrary bimodules,and prove the conjecture mentioned above.Obtaining this kind of vanishing results would be easy with a strong analogueof the Hochschild-Serre spectral sequence for Leibniz cohomology. Recall that theHochschild-Serre spectral sequence for a Lie algebra extension 0 → k → g → q → g by cochains whichvanish in case one inserts for a certain fixed number q elements of the ideal k in q arguments of the cochain (see [19, Sections 2 and 3]). When trying to generalizethis filtration from Lie algebras to Leibniz algebras, one needs to choose whether tofilter from the left or from the right. Another difficulty is that the arising spectralsequence does not converge to the cohomology of the Leibniz algebra, but rather tothe cohomology of some quotient complex. Furthermore, one must impose that theideal acts trivially from the left (right) on the left (right) Leibniz algebra. This lastissue excludes the application of the spectral sequence to many interesting idealsin the Leibniz algebra. Pirashvili [31, Theorem C] has constructed an analogue ofthe Hochschild-Serre spectral sequence using the filtration from the right for right Date : November 3, 2020.2010
Mathematics Subject Classification.
Primary 17A32; Secondary 17B56.
Key words and phrases.
Leibniz cohomology, Chevalley-Eilenberg cohomology, spectral se-quence, cohomological vanishing, invariant symmetric bilinear form, Cartan-Koszul map, com-plete Lie algebra, rigid Leibniz algebra, Witt algebra, Borel subalgebra, parabolic subalgebra,semi-simple Leibniz algebra, second Whitehead lemma, outer derivation.
Leibniz algebras and indicated how to use it together with a long exact sequencein order to extract cohomology. We use Pirashvili’s framework and extend thedual of his spectral sequence from symmetric bimodules to arbitrary bimodules(see Theorem 3.4). The two main changes of perspective with respect to [31] arethe systematic use of arbitrary bimodules and computations in which we considerground fields of all characteristics. We hope that this might be useful for furtherapplications in the future.The main application of Theorem 3.4 is Theorem 4.3 in which we compute thecohomology of a finite-dimensional semi-simple Leibniz algebra over a field of char-acteristic zero with coefficients in an arbitrary finite-dimensional bimodule. Thecase n = 2 of Theorem 4.3 is the second Whitehead lemma for Leibniz algebras.But note that contrary to Chevalley-Eilenberg cohomology, Leibniz cohomologyvanishes in any degree n ≥
2. This is one of several instances that we found byour computations in this paper which indicates that Leibniz cohomology behavesmore uniformly than Chevalley-Eilenberg cohomology. We also show by examplesthat the theorem fails in prime characteristic or for infinite-dimensional modules(see Examples E and F, respectively).As an immediate consequence of Theorem 4.3 we obtain the rigidity of finite-dimensional semi-simple Leibniz algebras in characteristic zero (see Corollary 4.7).More generally, we obtain a complete description of the cohomology of a finite-dimensional semi-simple left Leibniz algebra with coefficients in the adjoint bimod-ule and its (anti-)symmetric counterparts (see Theorem 4.5). In particular, wededuce that a finite-dimensional semi-simple non-Lie Leibniz algebra in character-istic zero always possesses outer derivations (see Corollary 4.6) which might besomewhat surprising as this shows that derivations of non-Lie Leibniz algebras aremore complicated than derivations of Lie algebras.In addition to the results just mentioned, we dualize another spectral sequenceobtained by Pirashvili for Leibniz homology (see [31, Theorem A]) that relates theLeibniz cohomology of a Lie algebra to its Chevalley-Eilenberg cohomology (seeTheorems 2.5 and 2.6). As an application we generalize some known results onrigidity to complete Lie algebras (see Corollary 2.9 and Corollary 2.10) and toparabolic subalgebras of finite-dimensional semi-simple Lie algebras (see Propo-sition 2.11). Moreover, we compute the Leibniz cohomology for the non-abeliantwo-dimensional Lie algebra (see Example A) and the three-dimensional Heisen-berg algebra (see Example B) with coefficients in irreducible Leibniz bimodules.The authors believe that Leibniz cohomology is an important invariant of a Liealgebra that behaves more uniformly than Chevalley-Eilenberg cohomology. Themotivation for including so many details in Section 2 was to provide the reader witha solid foundation for computing this invariant in arbitrary characteristics.The subject of Leibniz algebras, and especially its (co)homology theory, owes agreat deal to Jean-Louis Loday and Teimuraz Pirashvili (see [25], [24], [26], [31],and [27]). Many fundamental definitions and tools are due to them. While Lodayand Pirashvili work with right Leibniz algebras, we work with left Leibniz algebras.Obviously, results for left Leibniz algebras are equivalent to the corresponding re-sults for right Leibniz algebras. For the convenience of the reader we shall indicatewhere to find the corresponding formulae for left Leibniz algebras, even when theyhave been invented in the framework of right Leibniz algebras and are due to Lodayand Pirashvili.
EIBNIZ COHOMOLOGY 3
Teimuraz Pirashvili spotted an error in a first version of this article (see [32])which we have subsequently corrected. The error is related to another Hochschild-Serre type spectral sequence (see Remark (c) after Corollary 3.5) which we haveremoved from the present version because its E -term is more involved than weoriginally thought.In this paper we will follow the notation used in [14]. All tensor products areover the relevant ground field and will be denoted by ⊗ . For a subset X of a vectorspace V over a field F we let h X i F be the subspace of V spanned by X . We willdenote the space of linear transformations from an F -vector space V to an F -vectorspace W by Hom F ( V, W ). In particular, V ∗ := Hom F ( V, F ) will be the space oflinear forms on a vector space V over a field F . Moreover, S ( V ) will denote thesymmetric square of a vector space V . Finally, the identity function on a set X will be denoted by id X , and the set { , , , . . . } of non-negative integers will bedenoted by N . 1. Preliminaries
In this section we recall some definitions and collect several results that will beuseful in the remainder of the paper.A left Leibniz algebra is an algebra L such that every left multiplication operator L x : L → L , y xy is a derivation. This is equivalent to the identity(1.1) x ( yz ) = ( xy ) z + y ( xz )for all x, y, z ∈ L , which in turn is equivalent to the identity(1.2) ( xy ) z = x ( yz ) − y ( xz )for all x, y, z ∈ L . We will call both identities the left Leibniz identity . There is asimilar definition of a right Leibniz algebra but in this paper we will only considerleft Leibniz algebras.Every left Leibniz algebra has an important ideal, its Leibniz kernel, that mea-sures how much the Leibniz algebra deviates from being a Lie algebra. Namely, let L be a left Leibniz algebra over a field F . ThenLeib( L ) := h x | x ∈ L i F is called the Leibniz kernel of L . The Leibniz kernel Leib( L ) is an abelian ideal of L , and Leib( L ) = L when L = 0 (see [14, Proposition 2.20]). Moreover, L is a Liealgebra if, and only if, Leib( L ) = 0. It follows from the left Leibniz identity (1.2)that Leib( L ) ⊆ C ℓ ( L ), where C ℓ ( L ) := { c ∈ L | ∀ x ∈ L : cx = 0 } denotes the leftcenter of L .By definition of the Leibniz kernel, L Lie := L / Leib( L ) is a Lie algebra which wecall the canonical Lie algebra associated to L . In fact, the Leibniz kernel is thesmallest ideal such that the corresponding factor algebra is a Lie algebra (see [14,Proposition 2.22]).Next, we will briefly discuss left modules and bimodules of left Leibniz algebras.Let L be a left Leibniz algebra over a field F . A left L -module is a vector space M over F with an F -bilinear left L -action L × M → M , ( x, m ) x · m such that(1.3) ( xy ) · m = x · ( y · m ) − y · ( x · m )is satisfied for every m ∈ M and all x, y ∈ L . J ¨ORG FELDVOSS AND FRIEDRICH WAGEMANN
By virtue of [14, Lemma 3.3], every left L -module is an L Lie -module, and viceversa. Therefore left Leibniz modules are sometimes called Lie modules. Con-sequently, many properties of left Leibniz modules follow from the correspondingproperties of modules for the canonical Lie algebra.The correct concept of a module for a left Leibniz algebra L is the notion of aLeibniz bimodule. An L -bimodule is a left L -module M with an F -bilinear right L -action M × L → M , ( m, x ) m · x such that(1.4) ( x · m ) · y = x · ( m · y ) − m · ( xy )and(1.5) ( m · x ) · y = m · ( xy ) − x · ( m · y )are satisfied for every m ∈ M and all x, y ∈ L . In fact, all three identities (1.3),(1.4), and (1.5) are instances of the left Leibniz identity, written down for the leftLeibniz algebra L ⊕ M which is considered as an abelian extension in the theory ofnon-associative algebras, where the element m coccurs on the right, in the middle,or on the left, respectively (for the details see [14, Section 3]).The usual definitions of the notions of sub-(bi)module , irreducibility , completereducibility , composition series , homomorphism , isomorphism , etc., hold for leftLeibniz modules and Leibniz bimodules.Let L be a left Leibniz algebra over a field F , and let M be an L -bimodule. Then M is said to be symmetric if m · x = − x · m for every x ∈ L and every m ∈ M , and M is said to be anti-symmetric if m · x = 0 for every x ∈ L and every m ∈ M . Wecall M := h x · m + m · x | x ∈ L , m ∈ M i F the anti-symmetric kernel of M . It is known that M is an anti-symmetric L -sub-bimodule of M (see [14, Proposition 3.12]) such that M sym := M/M is symmetric(see [14, Proposition 3.13]).Moreover, for any L -bimodule M we will need its space of right L -invariants M L := { m ∈ M | ∀ x ∈ L : m · x = 0 } and the annihilator Ann bi L ( M ) := { x ∈ L | ∀ m ∈ M : x · m = 0 = m · x } . Our first result will be useful in the proof of Theorem 4.2.
Lemma 1.1.
Let L be a left Leibniz algebra, and let M be an L -bimodule such that M L = 0 . Then M is symmetric. In particular, Leib( L ) ⊆ Ann bi L ( M ) .Proof. Since M is anti-symmetric, it follows from the hypothesis that M = M L ⊆ M L = 0 . Hence we obtain from the definition of M that M is symmetric. The second partis then an immediate consequence of [14, Lemma 3.10]. (cid:3) It is clear from the definition of M L that an L -bimodule M is anti-symmetricif, and only if, M L = M . We will use Lemma 1.1 to show that the symmetry ofnon-trivial irreducible Leibniz bimodules can also be characterized by the behaviorof their spaces of right invariants. As a preparation for this, we need to know thatthe latter space is a sub-bimodule. EIBNIZ COHOMOLOGY 5
Lemma 1.2.
Let L be a left Leibniz algebra, and let M be an L -bimodule. Then M L is a sub-bimodule of M .Proof. It follows from (1.4) that M L is invariant under the left action on M , andit follows from (1.5) that M L is invariant under the right action on M . (cid:3) Remark.
More generally, the proof of Lemma 1.2 shows that M I is an L -sub-bimodule of M for every left ideal I of L .Now we can characterize the symmetry of a non-trivial irreducible Leibniz bi-module by the vanishing of its space of right invariants. In particular, for non-trivialirreducible Leibniz bimodules we obtain the converse of Lemma 1.1. (Recall that anirreducible bimodule M is a bimodule that has exactly two sub-bimodules, namely,0 and M . In particular, an irreducible bimodule is by definition a non-zero vectorspace.) Corollary 1.3.
Let L be a left Leibniz algebra, and let M be an irreducible L -bimodule. Then M is symmetric with non-trivial L -action if, and only if, M L = 0 .Proof. Since M is irreducible, we obtain from Lemma 1.2 that M L = 0 or M L = M .Suppose first that M is symmetric with non-trivial L -action. Then we have that M L = 0. On the other hand, the converse immediately follows from Lemma 1.1. (cid:3) Recall that every left L -module M of a left Leibniz algebra L determines a uniquesymmetric L -bimodule structure on M by defining m · x := − x · m for every element m ∈ M and every element x ∈ L (see [14, Proposition 3.15 (b)]). We will denotethis symmetric L -bimodule by M s . Similarly, every left L -module M with trivialright action is an anti-symmetric L -bimodule (see [14, Proposition 3.15 (a)]). Wewill denote this module by M a . Note that for any irreducible left L -module M the L -bimodules M s and M a are irreducible, and every irreducible L -bimodule arisesin this way from an irreducible left L -module (see [27, p. 415]).Similar to the boundary map in [25] for the homology of a right Leibniz algebrawith coefficients in a right module one can also introduce a coboundary map e d • for the cohomology of a left Leibniz algebra with coefficients in a left module asfollows.Let L be a left Leibniz algebra over a field F , and let M be a left L -module.For any non-negative integer n set CL n ( L , M ) := Hom F ( L ⊗ n , M ) and consider thelinear transformation e d n : CL n ( L , M ) → CL n +1 ( L , M ) defined by( e d n f )( x , . . . , x n +1 ) := n +1 X i =1 ( − i +1 x i · f ( x , . . . , ˆ x i , . . . , x n +1 )+ X ≤ i Now let M be an L -bimodule and for any non-negative integer n consider thelinear transformation d n : CL n ( L , M ) → CL n +1 ( L , M ) defined by(d n f )( x , . . . , x n +1 ) := n X i =1 ( − i +1 x i · f ( x , . . . , ˆ x i , . . . , x n +1 )+ ( − n +1 f ( x , . . . , x n ) · x n +1 + X ≤ i Let L be a left Leibniz algebra over a field F , and let M be a left L -module. Then the following statements hold: (a) If M is considered as a symmetric L -bimodule M s , then HL n ( L , M s ) = f HL n ( L , M ) EIBNIZ COHOMOLOGY 7 for every integer n ≥ . (b) If M is considered as an anti-symmetric L -bimodule M a , then HL ( L , M a ) = M and HL n ( L , M a ) ∼ = f HL n − ( L , Hom F ( L , M )) = HL n − ( L , Hom F ( L , M ) s ) for every integer n ≥ , where Hom F ( L , M ) is a left L -module via ( x · f )( y ) := x · f ( y ) − f ( xy ) for every f ∈ Hom F ( L , M ) and any elements x, y ∈ L .Proof. By virtue of the computation before Lemma 1.4, we only need to prove part(b). Note that the first part of (b) is just [14, Corollary 4.2 (b)].First, we show that Hom F ( L , M ) is a left L -module via the given action. Let f ∈ Hom F ( L , M ) and x, y, z ∈ L be arbitrary. Then we obtain from the definingidentity of a left Leibniz module (1.3) and the left Leibniz identity (1.2) that(( xy ) · f )( z ) = ( xy ) · f ( z ) − f (( xy ) z )= x · ( y · f ( z )) − y · ( x · f ( z )) − f ( x ( yz )) + f ( y ( xz )) , and ( x · ( y · f ))( z ) = x · ( y · f )( z ) − ( y · f )( xz )= x · ( y · f ( z )) − x · f ( yz ) − y · f ( xz ) + f ( y ( xz )) , as well as( y · ( x · f ))( z ) = y · ( x · f )( z ) − ( x · f )( yz )= y · ( x · f ( z )) − y · f ( xz ) − x · f ( yz ) + f ( x ( yz )) . Hence (( xy ) · f )( z ) = ( x · ( y · f )( z ) − ( y · ( x · f )( z ) for every z ∈ L , or equivalently,( xy ) · f = x · ( y · f ) − y · ( x · f ).Now we will prove the second part of (b). Let n be any positive integer. Considerthe linear transformations ϕ n : CL n ( L , M ) → CL n − ( L , Hom F ( L , M )) defined by ϕ n ( f )( x , . . . , x n − )( x ) := f ( x , . . . , x n − , x ) for any elements x , . . . , x n − , x ∈ L and ψ n : CL n − ( L , Hom F ( L , M )) → CL n ( L , M ) defined by ψ n ( g )( x , . . . , x n − , x n ):= g ( x , . . . , x n − )( x n ) for any elements x , . . . , x n − , x n ∈ L . Then ϕ n and ψ n areinverses of each other. J ¨ORG FELDVOSS AND FRIEDRICH WAGEMANN Next, we will show that e d n − ◦ ϕ n = ϕ n +1 ◦ d n . Compute( e d n − ◦ ϕ n )( f )( x , . . . , x n )( x ) = e d n − ( ϕ n ( f ))( x , . . . , x n )( x )= n X i =1 ( − i +1 ( x i · ϕ n ( f ))( x , . . . , ˆ x i , . . . , x n )( x )+ X ≤ i 1. Hence ϕ n induces an isomorphism of vector spaces betweenHL n ( L , M ) and f HL n − ( L , Hom F ( L , M )) for every integer n ≥ 1. In order to seethe remainder of the assertion, apply part (a). (cid:3) EIBNIZ COHOMOLOGY 9 In the special case of the trivial one-dimensional Leibniz bimodule we obtainfrom Lemma 1.4 the following result which will be needed in Section 4 (see [25,Exercise E.10.6.1] for the analogous result in Leibniz homology). Corollary 1.5. Let L be a left Leibniz algebra over a field F . Then for every integer n ≥ there are isomorphisms HL n ( L , F ) ∼ = f HL n − ( L , L ∗ ) = HL n − ( L , ( L ∗ ) s ) of vector spaces, where L ∗ is a left L -module via ( x · f )( y ) := − f ( xy ) for everylinear form f ∈ L ∗ and any elements x, y ∈ L . Remark. Note that [20, Theorem 3.5] is an immediate consequence of the case n = 2 of Corollary 1.5 and [14, Corollary 4.4 (a)].2. A relation between Chevalley-Eilenberg cohomology and Leibnizcohomology for Lie algebras Let g be a Lie algebra, and let M be a left g -module that is also viewed asa symmetric Leibniz g -bimodule M s . In this section, we will investigate how theChevalley-Eilenberg cohomology H • ( g , M ) and the Leibniz cohomology HL • ( g , M s )are related. The tools set forth in this section have been developed by Pirashvili,and we follow the analogous treatment for homology given in [31] very closely.The Chevalley-Eilenberg cohomology of a Lie algebra g with trivial coefficientsis not isomorphic (up to a degree shift) to the Chevalley-Eilenberg cohomology of g with coadjoint coefficients as it is the case for Leibniz cohomology (see Corol-lary 1.5). Instead these cohomologies are only related by a long exact sequence (seeProposition 2.1). The cohomology measuring the deviation from such an isomor-phism will appear in a spectral sequence (see Theorem 2.5) which can be used torelate the Leibniz cohomology of a Lie algebra to its Chevalley-Eilenberg cohomol-ogy (see Proposition 2.2).The exterior product map m n : Λ n g ⊗ g → Λ n +1 g given on homogeneous tensorsby x ∧ . . . ∧ x n ⊗ x x ∧ . . . ∧ x n ∧ x induces a monomorphism m • : C • ( g , F )[ − ֒ → C • ( g , g ∗ ) , where C • ( g , F ) is the truncated cochain complexC ( g , F ) := 0 and C n ( g , F ) := C n ( g , F ) for every integer n > . The cochain complex CR • ( g ) is defined by CR • ( g ) := Coker( m • )[ − • ( g ) := Ker( m • )[1] (see [35, (1.2.8),p. 9] for the definition of the degree shift and see [31, Section 1] for the definitionof CR • ( g )). We will mainly use the cochain complex in our paper, but the chaincomplex appears in Theorem 2.5 and its proof. Observe that classes in CR n ( g )are represented by cochains of degree n + 1 with values in g ∗ , i.e., they have n + 2arguments. From the short exact sequence0 → C • ( g , F )[ − → C • ( g , g ∗ ) → CR • ( g )[1] → Proposition 2.1. For every Lie algebra g over a field F there is a long exactsequence → H ( g , F ) → H ( g , g ∗ ) → HR ( g ) → H ( g , F ) → H ( g , g ∗ ) → HR ( g ) → · · · and an isomorphism H ( g , F ) ∼ = H ( g , g ∗ ) . Remark. If we assume that the characteristic of the ground field F is not 2, thenHR ( g ) ∼ = [ S ( g ) ∗ ] g is the space of invariant symmetric bilinear forms on g (see[31, p. 403]). As a consequence, we obtain from Proposition 2.1 in the case thatchar( F ) = 2 the five-term exact sequence0 → H ( g , F ) → H ( g , g ∗ ) → [ S ( g ) ∗ ] g → H ( g , F ) → H ( g , g ∗ ) , which generalizes [13, Proposition 1.3 (1) & (3)]. Note that the map [ S ( g ) ∗ ] g → H ( g , F ) is the classical Cartan-Koszul map defined by ω ω + B ( g , F ), where ω ( x ∧ y ∧ z ) := ω ( xy, z ) for any elements x, y, z ∈ g (see [31, p. 403]).For a Lie algebra g and a left g -module M viewed as a symmetric Leibniz g -bimodule M s , we have a natural monomorphismC • ( g , M ) ֒ → CL • ( g , M s ) . The cokernel of this morphism is by definition (up to a shift in the degree) thecochain complex C • rel ( g , M ):C • rel ( g , M ) := Coker(C • ( g , M ) → CL • ( g , M s ))[ − . We therefore have another long exact sequence. (For the isomorphisms in degrees0 and 1 see [14, Corollary 4.2 (a)] and [14, Corollary 4.4 (a)], respectively.) Proposition 2.2. Let g be a Lie algebra, and let M be a left g -module consideredas a symmetric Leibniz g -bimodule M s . Then there is a long exact sequence → H ( g , M ) → HL ( g , M s ) → H ( g , M ) → H ( g , M ) → HL ( g , M s ) → H ( g , M ) → · · · and isomorphisms HL ( g , M s ) ∼ = H ( g , M ) , HL ( g , M s ) ∼ = H ( g , M ) . Remark. If we again assume that the characteristic of the ground field F is not2, it follows from Theorem 2.5 below in conjunction with the remark after Propo-sition 2.1 that H ( g , F ) ∼ = HR ( g ) ∼ = [ S ( g ) ∗ ] g is the space of invariant symmetricbilinear forms on g . So when char( F ) = 2, we obtain the five-term exact sequence0 → H ( g , F ) → HL ( g , F ) → [ S ( g ) ∗ ] g → H ( g , F ) → HL ( g , F )as a special case of Proposition 2.2 (cf. [20, Proposition 3.2] for fields of characteristiczero). Note that Corollary 1.5 implies that the second terms of the five-term exactsequences in Proposition 2.1 and in Proposition 2.2 for M := F are isomorphic,but the fifth terms are not necessarily isomorphic (see the remark after Example Abelow).Observe that as for CR n ( g ), representatives of classes in C n rel ( g , M ) have n + 2arguments. EIBNIZ COHOMOLOGY 11 On the quotient cochain complex C • rel ( g , M ) there is the following filtration F p C n rel ( g , M ) = { [ c ] ∈ C n rel ( g , M ) | c ( x , . . . , x n +2 ) = 0 if ∃ j ≤ p + 1 : x j − = x j } . Note that the condition is independent of the representative c of the class [ c ]. Thisdefines a finite decreasing filtration(2.1) F C n rel ( g , M ) = C n rel ( g , M ) ⊃ F C n rel ( g , M ) ⊃ · · · ⊃ F n +1 C n rel ( g , M ) = { } . Then we have the following result: Lemma 2.3. This filtration is compatible with the Leibniz coboundary map d • .Proof. The Leibniz coboundary map, acting on a cochain c , is an alternating sumof operators d ij ( c ), δ i ( c ) and ∂ ( c ), where d ij ( c ) is the term involving the product ofthe i -th and the j -th element, δ i ( c ) is the term involving the left action of the i -thelement, and ∂ ( c ) is the term involving the right action of the ( n + 1)-th element.As the bimodule is symmetric, the term involving the right action can be countedamong the terms involving the left actions.We have to show that d • ( F p C n rel ( g , M )) ⊆ F p C n +1rel ( g , M ). We thus considerthe different terms of d • ( c ) with two equal elements as arguments in the first p + 1positions and have to show that all terms are zero. For d ij ( c ) with i, j ≤ p + 1,the assertion is clear because either the two equal elements do not occur in theproduct, and then it is correct, or at least one of them occurs, and then from theproduct terms of the sum of d ij and d ij +1 (or d ij and d ij − ) we obtain an element x i x ⊗ x + x ⊗ x i x , which is a sum of symmetric elements thanks to x i x ⊗ x + x ⊗ x i x = ( x i x + x ) ⊗ ( x i x + x ) − x i x ⊗ x i x − x ⊗ x . Even more elementary, the assertion holds for d ij ( c ) with i, j ≥ p + 1. For d ij ( c )with i ≤ p + 1 and j ≥ p + 2, the assertion is clear in case x i is not one of the equalelements. In case it is, the two terms corresponding to the product action of thetwo equal elements cancel as they are equal and have different sign.For the action terms δ i ( c ) the reasoning is similar. In case i ≤ p + 1, either thetwo equal elements do not occur and the assertion holds, or both occur and canceleach other because of the alternating sign. For δ i ( c ) with i ≥ p + 2, the assertionis clear in any case. (cid:3) The lemma implies that there is a spectral sequence of a filtered cochain complexassociated to this filtration which thanks to (2.1) converges in the strong (i.e., finite)sense to H n rel ( g , M ).The next step is then to compute the 0-th term of this spectral sequence, i.e.,the associated graded vector space of the filtration E p,q := F p C p + q rel ( g , M ) / F p +1 C p + q rel ( g , M ) . Observe that F p C p + q rel ( g , M ) = { c ∈ CL p + q +2 ( g , M s ) | c ( x , . . . , x p + q +2 ) = 0 if ∃ j ≤ p +1 : x j − = x j } / C p + q +2 ( g , M ) . In the quotient space E p,q , the term C p + q +2 ( g , M ), by which both filtration spacesare divided, disappears.Observe that the filtration can be expressed as F p C p + q rel ( g , M ) = { [ c ] ∈ C p + q rel ( g , M ) | c | Ker( ⊗ p +1 g → Λ p +1 g ) ⊗ ( ⊗ q +1 g ) = 0 } . This is useful, because by elementary linear algebra, we have F ⊥ / G ⊥ = Hom F ( G/F, M ) , where F ⊥ := { f : E → M | f | F = 0 } and G ⊥ := { f : E → M | f | G = 0 } for F ⊆ G ⊆ E .In order to be able to find E p,q , we therefore have to computeKer( ⊗ p +2 g → Λ p +2 g ) ⊗ ( ⊗ q g ) / Ker( ⊗ p +1 g → Λ p +1 g ) ⊗ ( ⊗ q +1 g ) . By using the isomorphism (see the proof of Theorem A in [31])Ker( ⊗ p +2 g → Λ p +2 g ) / Ker( ⊗ p +2 g → Λ p +1 g ⊗ g ) ∼ = Ker(Λ p +1 g ⊗ g → Λ p +2 g )and by applying that Hom F and ⊗ are adjoint functors, we obtain that E p,q = Hom F (Ker(Λ p +1 g ⊗ g → Λ p +2 g ) ⊗ CL q ( g ) , M )= Hom F (Ker(Λ p +1 g ⊗ g → Λ p +2 g ) , Hom F (CL q ( g ) , M ))= Hom F (CR p ( g ) , CL q ( g , M )) , by definition of the chain complex CR • ( g ) as the kernel of m • (up to a degree shift).In the case of a finite-dimensional Lie algebra g , we can use the isomorphismHom F ( U ⊗ V, W ) ∼ = U ∗ ⊗ Hom F ( V, W ) to write the E -term as E p,q = [Ker(Λ p +1 g ⊗ g → Λ p +2 g )] ∗ ⊗ CL q ( g , M ) . In this particular case, one may observe that the first tensor factor is the kernel ofthe exterior multiplication map m • , and thusKer(Λ p +1 g ⊗ g → Λ p +2 g ) ∗ = Ker( m • )[1] ∗ = Coker( m • )[ − 1] = CR p ( g ) . Therefore the term E p,q takes in this particular case the form E p,q = CR p ( g ) ⊗ CL q ( g , M ) . Next, we will determine the differential on E p,q : Lemma 2.4. The differential d on E p,q ∼ = Hom F (CR p ( g ) , CL q ( g , M )) is inducedby the coboundary operator d • on C • rel ( g , M ) . More precisely, we have that d p,q ( f ) := d q CL q ( g ,M ) ◦ f for every linear transformation f ∈ Hom F (CR p ( g ) , CL q ( g , M )) .Proof. By definition, the differential d of the spectral sequence is the differentialwhich is induced by the Leibniz coboundary map d • on the associated gradedquotientsd : F p C p + q rel ( g , M ) / F p +1 C p + q rel ( g , M ) → F p C p + q +1rel ( g , M ) / F p +1 C p + q +1rel ( g , M ) . In order to examine which terms d ij ( c ), δ i ( c ) and ∂ ( c ) are zero for a cochain c ∈F p C p + q rel ( g , M ), we have to insert two consecutive equal elements in the argumentsof c within the first p + 2 arguments.Now, by the same reasoning as in the proof of Lemma 2.3, the terms d ij ( c ) vanishin case i, j ≤ p + 2, because in case the equal elements are not involved, the formulafor d ij ( c ) diminishes the number of arguments by one, and as c is of degree p in thefiltration, this then gives zero. In case the elements occur, they create once againa symmetric element of the form x i x ⊗ x + x ⊗ x i x . Also for d ij ( c ) with i ≤ p + 2and j ≥ p + 3, the terms are zero when the equal elements are not involved, andare zero in addition with d ij +1 ( c ) (or d ij − ( c )), in case of multiplying with one EIBNIZ COHOMOLOGY 13 of the equal elements. The terms δ i ( c ) for i ≤ p + 1 vanish as the correspondingformula diminishes the number of arguments by one in case the equal elements donot occur, and annihilate each other in case they occur.Thus, there remain the terms d ij ( c ) with i, j ≥ p +3, δ i ( c ) with i ≥ p +3, and ∂ ( c ),which form together the coboundary map of the cochain complex CL • ( g , M ). (cid:3) Consequently, in the general case we obtain for the first term of the spectralsequence: E p,q = Hom F (CR p ( g ) , HL q ( g , M s )) , and in the case of a finite-dimensional Lie algebra g we have that E p,q = CR p ( g ) ⊗ HL q ( g , M s )) . Now we proceed to identify the differential d on E p,q . The differential d is stillinduced by the Leibniz coboundary map on the filtered cochain complex. As theclasses in HL q ( g , M s ) are represented by cocycles, the part of the Leibniz cobound-ary operator d q CL q ( g ,M s ) constituting the differential must be zero. The action of oneof the remaining terms on HL q ( g , M s ) must also be zero since the Cartan relationsfor Leibniz cohomology (due to Loday and Pirashvili [26, Proposition 3.1]) implythat a Leibniz algebra acts trivially on its cohomology. (For the reader interestedin left Leibniz algebras, a proof of these formulas adapted to this case can be foundin [10, Proposition 1.3.2].) Note that the Cartan relations only hold for q ≥ 1. Butas the Leibniz g -bimodule M is symmetric, the action of g on HL ( g , M s ) is alsotrivial. Therefore, the remaining terms of the differential constitute the coboundaryoperator d p CL p ( g ) on CR p ( g ).Consequently, in the general case we obtain for the second term of the spectralsequence: E p,q = HR p ( g , HL q ( g , M s )) , where the right-hand side denotes the HR-cohomology with values in the trivial g -module HL • ( g , M s ). It is the cohomology of the cochain complex arising fromapplying the exact functor Hom F ( − , HL • ( g , M s )) to the chain complex CR • ( g ). Itfollows from the exactness of this functor and the Universal Coefficient Theorem(for example, see [35, Theorem 3.6.5]) that we can express the E -term as E p,q = Hom F (HR p ( g ) , HL q ( g , M s )) , and in the special case of a finite-dimensional Lie algebra g , we have that E p,q = HR p ( g ) ⊗ HL q ( g , M s ) . This discussion proves the following result which (up to dualization) is Theo-rem A in [31]. In the case of trivial coefficients (and possibly topological Fr´echetLie algebras) the second part of Theorem 2.5 has also been obtained by Lodder (see[28, Theorem 2.10]). Theorem 2.5. Let g be a Lie algebra, and let M be a left g -module considered asa symmetric Leibniz g -bimodule M s . Then there is a spectral sequence convergingto H • rel ( g , M ) with second term E p,q = Hom F (HR p ( g ) , HL q ( g , M s )) . Moreover, if g is finite dimensional, then the E -term of this spectral sequence canbe written as E p,q = HR p ( g ) ⊗ HL q ( g , M s )) . Remark. As the spectral sequence of Theorem 2.5 is the spectral sequence of afiltered cochain complex, the higher differentials in this spectral sequence are againinduced by the Leibniz coboundary operator d • . We will see in Example B belowan instance of a concrete computation of the differential d .Our main application of the spectral sequence will be the next theorem which isa refinement of the cohomological analogue of [31, Corollary 1.3]: Theorem 2.6. Let g be a Lie algebra, let M be a left g -module considered as a sym-metric Leibniz g -bimodule M s , and let n be a non-negative integer. If H k ( g , M ) = 0 for every integer k with ≤ k ≤ n , then HL k ( g , M s ) = 0 for every integer k with ≤ k ≤ n and HL n +1 ( g , M s ) ∼ = H n +1 ( g , M ) as well as HL n +2 ( g , M s ) ∼ = H n +2 ( g , M ) .In particular, H • ( g , M ) = 0 implies that HL • ( g , M s ) = 0 .Proof. The proof follows the proof of Corollary 1.3 in [31] very closely.According to Proposition 2.2, it suffices to prove that H k ( g , M ) = 0 for everyinteger k with 0 ≤ k ≤ n implies that H n rel ( g , M ) = 0 for every integer k with 0 ≤ k ≤ n . We proceed by induction on n . In the case n = 0, the hypothesis yields that E , = 0 for the second term of the spectral sequence of Theorem 2.5, and thereforewe obtain from the convergence of the spectral sequence that H ( g , M ) = 0 whichinitializes the induction.So suppose now that n ≥ k ( g , M s ) = 0 for every integer k with 0 ≤ k ≤ n + 1. By induction hypothesis, we obtain that H n rel ( g , M ) = 0 for every integer k with 0 ≤ k ≤ n . Hence it follows from Proposition 2.2 that HL k ( g , M s ) = 0 for everyinteger k with 0 ≤ k ≤ n and HL n +1 ( g , M s ) ∼ = H n +1 ( g , M ) = 0. Consequently, thesecond term E p,q of the spectral sequence in Theorem 2.5 is zero for p + q ≤ n + 1,and therefore H n +1rel ( g , M ) = 0.Finally, the isomorphisms in degree n +1 and n +2, respectively, are an immediateconsequence of Proposition 2.2. (cid:3) Remark. Note that the converse of Theorem 2.6 is also true, namely, H k ( g , M ) = 0for every integer k with 0 ≤ k ≤ n if, and only if, HL k ( g , M s ) = 0 for every integer k with 0 ≤ k ≤ n . In particular, H • ( g , M ) = 0 if, and only if, HL • ( g , M s ) = 0.Next, we illustrate the use of the spectral sequence of Theorem 2.5 and theassociated long exact sequences (see Propositions 2.1 and 2.2) by two examples.We begin by computing the Leibniz cohomology of the smallest non-nilpotent Liealgebra with coefficients in an arbitrary irreducible Leibniz bimodule (see also [31,Example 1.4 i)] for trivial coeffcients in characteristic = 2). Note that for a groundfield of characteristic 2 the Leibniz cohomology of this Lie algebra is far morecomplicated than for a field of characteristic = 2.In the case that the irreducible Leibniz bimodule is of finite dimension = 1 wecan prove more generally for an arbitrary supersolvable Lie algebra the followingvanishing result. Proposition 2.7. Let g be a finite-dimensional supersolvable Lie algebra over afield F , and let M be a finite-dimensional irreducible Leibniz g -bimodule such that dim F M = 1 . Then HL n ( g , M ) = 0 for every positive integer n . Moreover, if M issymmetric, then HL n ( g , M ) = 0 for every non-negative integer n .Proof. If M is symmetric, then the assertion is an immediate consequence of The-orem 2.6 in conjunction with [3, Theorem 3]. EIBNIZ COHOMOLOGY 15 Now suppose that M is not symmetric. Then it follows from [14, Theorem 3.14]that M is anti-symmetric. We obtain from Lemma 1.4 (b) thatHL n ( g , M ) ∼ = HL n − ( g , Hom F ( g , M ) s ) ∼ = HL n − ( g , ( g ∗ ⊗ M ) s )for every positive integer n . By definition of supersolvability, the adjoint g -modulehas a composition series g ad ,ℓ = g n ⊃ g n − ⊃ · · · ⊃ g ⊃ g = 0such that dim F g j / g j − = 1 for every integer 1 ≤ j ≤ n . From the short exactsequences 0 → g j − → g j → g j / g j − → 0, we obtain by dualizing, tensoring eachterm with M , and symmetrizing the short exact sequences:0 → [( g j / g j − ) ∗ ⊗ M ] s → ( g ∗ j ⊗ M ) s → ( g ∗ j − ⊗ M ) s → ≤ j ≤ n . Since M is irreducible and dim F g j / g j − = 1, weconclude that [( g j / g j − ) ∗ ⊗ M ] s is an irreducible symmetric Leibniz g -bimodule.Moreover, we have that dim F [( g j / g j − ) ∗ ⊗ M ] s = 1 as dim F M = 1. Hence weobtain inductively from the long exact cohomology sequence that HL n ( g , M ) ∼ =HL n − ( g , ( g ∗ ⊗ M ) s ) = 0 for every positive integer n . (cid:3) Remark. It follows from Lie’s theorem that every finite-dimensional irreducibleLeibniz bimodule of a finite-dimensional solvable Lie algebra over an algebraicallyclosed field of characteristic zero is one-dimensional (see [22, Corollary 4.1 A]).Consequently, in this case the hypothesis of Proposition 2.7 is never satisfied, andthus this result is only applicable over non-algebraically closed fields of characteristiczero or over fields of prime characteristic.By virtue of Proposition 2.7, in the next example it is enough to consider one-dimensional Leibniz bimodules. Example A. Let F denote an arbitrary field, and let a := F h ⊕ F e be the non-abelian two-dimensional Lie algebra over F with multiplication determined by he = e = − eh . For any scalar λ ∈ F one can define a one-dimensional left a -module F λ := F λ with a -action defined by h · λ := λ λ and e · λ := 0. Then theChevalley-Eilenberg cohomology of a with coefficients in F λ is as follows:H n ( a , F λ ) ∼ = (cid:26) F if λ = 0 and n = 0 , λ = 1 and n = 1 , 20 otherwise . In particular, if λ = 0 , 1, then H • ( a , F λ ) = 0.First, let us consider F λ as a symmetric Leibniz a -bimodule ( F λ ) s . Then itfollows from Theorem 2.6 that HL • ( a , ( F λ ) s ) = 0 for λ = 0 , λ = 0 , 1, and forthe anti-symmetric Leibniz a -bimodules ( F λ ) a , let M be an arbitrary left a -moduleconsidered as a symmetric Leibniz a -bimodule M s . Since H n ( a , M ) = 0 for everyinteger n ≥ 3, we obtain from Proposition 2.2 the short exact sequence(2.2) 0 → H ( a , M ) → HL ( a , M s ) → H ( a , M ) → n ( a , M s ) ∼ = H n − ( a , M ) for every integer n ≥ . Moreover, we have that HL ( a , M s ) ∼ = M a and HL ( a , M s ) ∼ = H ( a , M ). For the computation of the relative cohomology spaces H n rel ( a , M ) we need thecoadjoint Chevalley-Eilenberg cohomology of a . It is easy to verify thatdim F H ( a , a ∗ ) = 1 , dim F H ( a , a ∗ ) = (cid:26) F ) = 21 if char( F ) = 2 , and dim F H ( a , a ∗ ) = (cid:26) F ) = 20 if char( F ) = 2 . Consequently, we have to consider the cases char( F ) = 2 and char( F ) = 2 differently.Let us first assume that char( F ) = 2. Then it follows from Proposition 2.1 thatHR ( a ) ∼ = H ( a , a ∗ ) ∼ = F and HR n ( a ) = 0 for every integer n ≥ 1. Hence we derivefrom the spectral sequence of Theorem 2.5 that H n rel ( a , M ) ∼ = HL n ( a , M s ) for everynon-negative integer n . In conclusion, we obtain from (2.2) and (2.3) that(2.4) HL ( a , M s ) ∼ = M a ⊕ H ( a , M )and(2.5) HL n ( a , M s ) ∼ = HL n − ( a , M s ) for every integer n ≥ . As an immediate consequence, in the case of char( F ) = 2 we deduce thatdim F HL n ( a , F ) = 1 for every non-negative integer n anddim F HL n ( a , ( F ) s ) = (cid:26) n = 01 if n > . In summary, we have for the Leibniz cohomology of a over a field F of char-acteristic = 2 with coefficients in a one-dimensional symmetric Leibniz bimodulethatdim F HL n ( a , ( F λ ) s ) = (cid:26) λ = 0 and n is arbitrary or if λ = 1 and n > 00 otherwise . Next, let us assume that char( F ) = 2. Then it follows from Proposition 2.1 thatHR ( a ) ∼ = H ( a , a ∗ ) ∼ = F , HR ( a ) ∼ = H ( a , a ∗ ) ∼ = F , and HR n ( a ) = 0 for everyinteger n ≥ 2. Hence in the spectral sequence of Theorem 2.5, we have only twonon-zero columns, namely the p = 0 and the p = 1 column. In the p = 0 column,we have spaces F ⊗ HL q ( a , M s ) ∼ = HL q ( a , M s ) ⊕ HL q ( a , M s ), while in the p = 1column, we have just HL q ( a , M s ) for every integer q ≥ 0. Therefore, the spectralsequence degenerates at the term E , and for every integer n ≥ n rel ( a , M ) ∼ = HL n ( a , M s ) ⊕ HL n ( a , M s ) ⊕ HL n − ( a , M s ) , and H ( a , M ) ∼ = E , ∼ = HL ( a , M s ) ⊕ HL ( a , M s ) ∼ = M a ⊕ M a . This, together with (2.2), (2.3), and induction yields the recursive relation(2.6) HL n ( a , M s ) ∼ = HL n − ( a , M s ) ⊕ HL n − ( a , M s ) for every integer n ≥ . As a consequence, we obtain for the Leibniz cohomology of a over a field F ofcharacteristic 2 with coefficients in a one-dimensional symmetric Leibniz bimodule EIBNIZ COHOMOLOGY 17 that dim F HL n ( a , ( F λ ) s ) = f n +1 if λ = 0 f n if λ = 10 otherwisefor every non-negative integer n , where f n denotes the n -th term of the standardFibonacci sequence given by f := 0, f := 1, and f n := f n − + f n − for everyinteger n ≥ 2. In particular, we have thatHL n ( a , ( F ) s ) ∼ = HL n − ( a , F ) for every integer n ≥ . Next, let us consider F λ as an anti-symmetric Leibniz a -bimodule ( F λ ) a withthe same left a -action as above and with the trivial right a -action (see Section 1).Then we conclude from Lemma 1.4 (b) thatdim F HL ( a , ( F λ ) a ) = 1 for every λ ∈ F . Let us now compute HL n ( a , ( F λ ) a ) for any integer n ≥ 1. It follows fromLemma 1.4 (b) that(2.7) HL n ( a , ( F λ ) a ) ∼ = HL n − ( a , Hom F ( a , F λ ) s ) ∼ = HL n − ( a , ( a ∗ ⊗ F λ ) s ) . A straightforward computation shows that0 → F λ → a ∗ ⊗ F λ → F λ − → a -modules. Then we obtain from the long exactcohomology sequence and another straightforward computation in the case λ = 1:dim F ( a ∗ ⊗ F λ ) a = (cid:26) λ = 00 otherwise , dim F H ( a , a ∗ ⊗ F λ ) = λ = 0 and char( F ) = 21 if λ = 0 , F ) = 2 , F H ( a , a ∗ ⊗ F λ ) = (cid:26) λ = 0 and char( F ) = 2 or λ = 2 and char( F ) = 20 otherwise . If char( F ) = 2, we conclude by applying (2.7) and (2.4) to the symmetric Leibniz a -bimodule M s := ( a ∗ ⊗ F λ ) s thatdim F HL ( a , ( F λ ) a ) = (cid:26) λ = 00 otherwise , and dim F HL ( a , ( F λ ) a ) = dim F HL ( a , ( F λ ) a ) = (cid:26) λ = 0 , 20 otherwise , Finally, we use (2.5) to deduce for every integer n ≥ F HL n ( a , ( F λ ) a ) = (cid:26) λ = 0 , 20 otherwise . In summary, we have for the Leibniz cohomology of a over a field F of charac-teristic = 2 with coefficients in a one-dimensional anti-symmetric Leibniz bimodulethatdim F HL n ( a , ( F λ ) a ) = λ = 0 and n is arbitrary or λ = 2 and n ≥ n = 0 and λ is arbitrary0 otherwise . If char( F ) = 2, we obtain by applying (2.7) and (2.6):dim F HL n ( a , ( F λ ) a ) = n = 0 and λ is arbitrary f n +1 if λ = 0 and n is arbitrary0 otherwise . Remark. Since every invariant symmetric bilinear form on a is a multiple of theKilling form, we have that [ S ( a ) ∗ ] a ∼ = F . On the other hand, from the computationsin Example A we obtain when char( F ) = 2 thatH ( a , F ) ∼ = HR ( a ) ∼ = H ( a , a ∗ ) ∼ = F . This shows that, in general, HR ( a ) = [ S ( a ) ∗ ] a and H ( a , F ) = [ S ( a ) ∗ ] a .Moreover, the computations in Example A show that dim F HL ( a , F ) = 3, butdim F H ( a , a ∗ ) ≤ 1. Hence the fifth terms of the five-term exact sequences inProposition 2.1 and in Proposition 2.2 for M := F are not necessarily isomorphic.Similar to Proposition 2.7, we have the following general vanishing result: Proposition 2.8. Let g be a finite-dimensional nilpotent Lie algebra, and let M bea finite-dimensional non-trivial irreducible Leibniz g -bimodule. Then HL n ( g , M ) =0 for every positive integer n . Moreover, if M is symmetric, then HL n ( g , M ) = 0 for every non-negative integer n .Proof. If M is symmetric, then the assertion is an immediate consequence of The-orem 2.6 and [3, Lemma 3].According to [14, Theorem 3.14], we can suppose that M is anti-symmetric. Weobtain from Lemma 1.4 (b) thatHL n ( g , M ) ∼ = HL n − ( g , Hom F ( g , M ) s ) ∼ = HL n − ( g , ( g ∗ ⊗ M ) s )for every positive integer n . By refining the descending central series of g , one canconstruct a composition series g ad ,ℓ = g n ⊃ g n − ⊃ · · · ⊃ g ⊃ g = 0of the adjoint g -module such that g j / g j − is the trivial one-dimensional g -module F for every integer 1 ≤ j ≤ n . From the short exact sequences0 → g j − → g j → F → , we obtain by dualizing, tensoring each term with M , and symmetrizing the shortexact sequences: 0 → M s → ( g ∗ j ⊗ M ) s → ( g ∗ j − ⊗ M ) s → ≤ j ≤ n . Since M is a non-trivial irreducible left g -module, weconclude that M s is a non-trivial irreducible symmetric Leibniz g -bimodule. Hencewe obtain inductively from the long exact cohomology sequence that HL n ( g , M ) ∼ =HL n − ( g , ( g ∗ ⊗ M ) s ) = 0 for every positive integer n . (cid:3) EIBNIZ COHOMOLOGY 19 Since the Leibniz cohomology of an abelian Lie algebra with trivial coefficientsis known, in Example B we compute this cohomology for the smallest non-abeliannilpotent Lie algebra. Note that in [31, Example 1.4. iv)] the corresponding Leib-niz homology has been computed. In fact, homology and cohomology of a finite-dimensional Leibniz algebra L with trivial coefficients are isomorphic, as we havethe duality isomorphism CL • ( L , F ) ∗ ∼ = CL • ( L , F ) already on the level of cochaincomplexes. Therefore our results coincide with those of Pirashvili. We furthermorecompute in Example C the Leibniz cohomology of the smallest nilpotent non-LieLeibniz algebra with coefficients in the trivial Leibniz bimodule. Example B. Let F denote an arbitrary field of characteristic = 2, and let h := F x ⊕ F y ⊕ F z be the three-dimensional Heisenberg algebra over F with multiplicationdetermined by xy = z = − yx . Then the Chevalley-Eilenberg cohomology of h withcoefficients in the trivial module F is well-known:dim F H n ( h , F ) = n = 0 , 32 if n = 1 , . n ≥ F HL ( h , F ) = 1 and dim F HL ( h , F ) = 2.As H n ( h , F ) = 0 for every integer n ≥ 4, we obtain from Proposition 2.2 thefollowing six-term exact sequence:0 → H ( h , F ) → HL ( h , F ) → H ( h , F ) → H ( h , F ) → HL ( h , F ) → H ( h , F ) → n ( h , F ) ∼ = H n − ( h , F ) for every integer n ≥ . Since we assume that char( F ) = 2, it follows from the remark after Proposi-tion 2.2 that we can identify H ( h , F ) with the space of invariant symmetric bilin-ear forms on h and the map H ( h , F ) → H ( h , F ) with the classical Cartan-Koszulmap. It is easy to see that the latter map is zero for the Heisenberg algebra, whichyields the surjectivity of the map HL ( h , F ) → H ( h , F ) and the injectivity of themap H ( h , F ) → HL ( h , F ). As a consequence, we obtain the following two shortexact sequences: 0 → H ( h , F ) → HL ( h , F ) → H ( h , F ) → , → H ( h , F ) → HL ( h , F ) → H ( h , F ) → . In order to compute H ( h , F ) and H ( h , F ), we need the coadjoint Chevalley-Eilenberg cohomology of h . We have that dim F H ( h , h ∗ ) = 2, dim F H ( h , h ∗ ) = 5,dim F H ( h , h ∗ ) = 4, and H ( h , h ∗ ) = 1. These dimensions can be computed directly,but for the complex numbers as a ground field it also follows from the main resultof [29] in conjunction with Poincar´e duality (for the latter see [36, Theorem 3.4]).Similar to the discussion of the consequences of Proposition 2.2 above, we obtainfrom Proposition 2.1 the two short exact sequences0 → H ( h , F ) → H ( h , h ∗ ) → HR ( h ) → , → H ( h , F ) → H ( h , h ∗ ) → HR ( h ) → , the isomorphism HR ( h ) ∼ = H ( h , h ∗ ), and HR n ( h ) = 0 for every integer n ≥ F HR ( h ) = dim F H ( h , h ∗ ) − dim F H ( h , F ) = 5 − , dim F HR ( h ) = dim F H ( h , h ∗ ) − dim F H ( h , F ) = 4 − , and dim F HR ( h ) = dim F H ( h , h ∗ ) = 1 , respectively. Therefore we obtain from H ( h , F ) ∼ = HR ( h ) thatdim F HL ( h , F ) = dim F H ( h , F ) + dim F H ( h , F ) = 2 + 3 = 5 . Now we want to apply the spectral sequence of Theorem 2.5. For this let uscompute the differentiald , : E , = HR ( h ) ⊗ HL ( h , F ) → E , = HR ( h ) ⊗ HL ( h , F ) . In characteristic = 2, an element of HR ( h ) is an invariant symmetric bilinear form ω . It is considered as a 1-cochain with values in h ∗ and, as it is a representativeof an element of a quotient cochain complex, it is zero in case it is skew-symmetricin all entries. Take furthermore a cocycle c ∈ CL ( h , F ) and compute for threeelements r, s, t ∈ h :d ( ω ⊗ c )( r, s, t ) = ω ( rs, − ) c ( t ) + ω ( s, − ) c ( rt ) − ω ( r, − ) c ( st ) ++ ω ( s, r − ) c ( t ) − ω ( r, s − ) c ( t ) + ω ( r, t − ) c ( s )Now as c is a cocycle with trivial coefficients, c vanishes on products, thus thesecond and third terms are zero. Furthermore, the first and fourth term cancel bythe invariance of the form and skew-symmetry of the Lie product. We are left withthe two last terms − ω ( r, s − ) c ( t ) + ω ( r, t − ) c ( s ), which are skew-symmetric in thethree entries of the element in HR ( h ) and vanish therefore as well. In conclusion,the differential d , is zero, and we have thatH ( h ) = HR ( h ) ⊗ HL ( h , F ) ⊕ HR ( h ) ⊗ HL ( h , F ) . This implies in turndim F HL ( h , F ) = dim F H ( h , F ) + dim F H ( h , F ) = 1 + 9 = 10 . It seems that all differentials d are zero and thus that this scheme persiststo yield the dimensions of the higher H n rel ( h , F ) and thus of HL n ( h , F ) (see thedimension formula in [31, Example 1.4 iv)]). Remark. As a by-product of the above computations we obtain that the space[ S ( h ) ∗ ] h of invariant symmetric bilinear forms on h is three-dimensional whenchar( F ) = 2.We proceed by proving an extension of a result by Fialowski, Magnin, and Man-dal (see Corollary 2 in [16]), namely, the fact that the vanishing of the center C ( g )of a Lie algebra g implies HL ( g , g ad ) = H ( g , g ), where g ad denotes the adjointLeibniz g -bimodule induced by the left and right multiplication operator. Notethat for Lie algebras, this bimodule is indeed symmetric.Moreover, we observe that H ( g , g ) = C ( g ). Therefore, it is an immediate con-sequence of the case n = 0 of Theorem 2.6 that the vanishing of the center impliesHL ( g , g ad ) = H ( g , g ). By the same token for n = 1, we can extend this result tocomplete Lie algebras, i.e., to those Lie algebras g for which H ( g , g ) = H ( g , g ) = 0: Corollary 2.9. Let g be a complete Lie algebra. Then HL ( g , g ad ) ∼ = H ( g , g ) and HL ( g , g ad ) ∼ = H ( g , g ) . EIBNIZ COHOMOLOGY 21 A class of examples of complete Lie algebras over an algebraically closed field F of characteristic zero consists of those finite-dimensional Lie algebras g for which g has the same dimension as its Lie algebra of derivations and dim F g / g > n = 3 of Theorem 2.6 the followingresult: Corollary 2.10. Let W := Der( F [ t, t − ]) be the two-sided Witt algebra over a field F of characteristic zero. Then HL ( W , W ad ) = 0 and HL ( W , W ad ) = 0 . Moreover, HL ( W , W ad ) ∼ = H ( W , W ) and HL ( W , W ad ) ∼ = H ( W , W ) . Remark. Very recently, Camacho, Omirov, and Kurbanbaev also proved that thesecond adjoint Leibniz cohomology of W vanishes (see [7, Theorem 4]) by explicitlyshowing that every adjoint Leibniz 2-cocycle (resp. Leibniz 2-coboundary) is anadjoint Chevalley-Eilenberg 2-cocycle (resp. Chevalley-Eilenberg 2-coboundary) for W .We conclude this section with another application of Theorem 2.6. Let F be analgebraically closed field of characteristic zero, let n be a non-negative integer, andlet L n ( F ) ⊆ F n denote the affine variety of structure constants of the n -dimensionalleft Leibniz algebras over F with respect to a fixed basis of F n . Then the generallinear group GL n ( F ) acts on L n ( F ), and a point (= Leibniz multiplication law) φ ∈ L n ( F ) is called rigid if the orbit GL n ( F ) · φ is open in L n ( F ). It follows fromCorollary 2.10 in conjunction with [2, Th´eor`eme 3] that the infinite-dimensionaltwo-sided Witt algebra over an algebraically closed field of characteristic zero isrigid as a Leibniz algebra.It is well known that the Chevalley-Eilenberg cohomology of the non-abelian two-dimensional Lie algebra with coefficients in the adjoint module vanishes. Accordingto Theorem 2.6, this implies that the corresponding Leibniz cohomology vanishes aswell. Note that the non-abelian two-dimensional Lie algebra is the standard Borelsubalgebra of the split three-dimensional simple Lie algebra sl .Similarly, by applying Theorem 2.6 in conjunction with [34, Theorem 1] (seealso [23, Section 1]) one obtains the following more general result in characteristiczero (cf. also [31, Proposition 2.3] for the rigidity of parabolic subalgebras). Recallthat a subalgebra of a semi-simple Lie algebra g is called parabolic if it contains amaximal solvable (= Borel) subalgebra of g . Proposition 2.11. Let p be a parabolic subalgebra of a finite-dimensional semi-simple Lie algebra over a field of characteristic zero. Then HL n ( p , p ad ) = 0 for everynon-negative integer n . In particular, parabolic subalgebras of a finite-dimensionalsemi-simple Lie algebra over an algebraically closed field of characteristic zero arerigid as Leibniz algebras. Remark. It would be interesting to know whether Proposition 2.11 remains validin prime characteristic. A Hochschild-Serre type spectral sequence for Leibnizcohomology In this section we consider a Leibniz analogue of the Hochschild-Serre spectralsequence for the Chevalley-Eilenberg cohomology of Lie algebras that convergesto some relative cohomology. It will play a predominant role in Section 4. Thehomology version of this spectral sequence with values in symmetric bimodulesis due to Pirashvili (see [31, Theorem C]). Our arguments follow Pirashvili veryclosely, but we include all the details as it turns out that the spectral sequenceholds more generally for arbitrary bimodules.Let π : L → Q be an epimorphism of left Leibniz algebras, and let M be a Q -bimodule. Then M is also an L -bimodule via π . Moreover, the epimorphisms π ⊗ n : L ⊗ n → Q ⊗ n induce a monomorphism CL • ( Q , M ) → CL • ( L , M ) of cochaincomplexes. Now setCL • ( L | Q , M ) := Coker(CL • ( Q , M ) → CL • ( L , M ))[ − • ( L | Q , M ) := H • (CL • ( L | Q , M )) . Then by applying the long exact cohomology sequence to the short exact sequence0 → CL • ( Q , M ) → CL • ( L , M ) → CL • ( L | Q , M )[1] → Proposition 3.1. For every epimorphism π : L → Q of left Leibniz algebras andevery Q -bimodule M there exists a long exact sequence → HL ( Q , M ) → HL ( L , M ) → HL ( L | Q , M ) → HL ( Q , M ) → HL ( L , M ) → HL ( L | Q , M ) → · · · Let us now derive Pirashvili’s analogue of the Hochschild-Serre spectral sequencefor Leibniz cohomology (see [31, Theorem C] for the homology version). WhilePirashvili considers only symmetric bimodules, we extend the dual of his spectralsequence to arbitrary bimodules. The construction of this spectral sequence is verysimilar to the construction of the spectral sequence in Theorem 2.5.We consider the following filtration on the cochain complex CL • ( L , M )[ − F p CL n ( L , M )[ − 1] := { c ∈ CL n +1 ( L , M ) | c ( x , . . . , x n +1 ) = 0 if ∃ i ≤ p : x i ∈ I } . This defines a finite decreasing filtration(3.1) F CL n ( L , M )[ − 1] = CL n ( L , M )[ − ⊃ F CL n ( L , M )[ − ⊃ · · ·· · · ⊃ F n +1 CL n ( L , M )[ − 1] = CL n ( Q , M )[ − , Then we have the following result: Lemma 3.2. This filtration is compatible with the Leibniz coboundary map d • .Proof. We have to prove that d • ( F p CL n ( L , M )[ − ⊆ F p CL n +1 ( L , M )[ − ij ( c ), δ i ( c ), and ∂ ( c ), which constitute d ( c ),where we have inserted an element of I within the first p arguments. The vanishingis clear for the terms d ij ( c ) with i, j ≤ p , because even if the element of I occursin the product, the product will again be in the ideal I . The vanishing is also clear EIBNIZ COHOMOLOGY 23 for the terms d ij ( c ) with i, j ≥ p + 1. Concerning the terms d ij ( c ) with i ≤ p and j ≥ p + 1, we use the condition I ⊆ C ℓ ( L ) to conclude that these are zero.The action terms follow a similar pattern. The terms δ i ( c ) with i ≤ p vanish,because either the element of I occurs in the arguments, or it acts on M , which iszero by assumption. The terms δ i ( c ) with i ≥ p + 1 are zero for elementary reasons,as is the term ∂ ( c ). (cid:3) Thanks to (3.1), the associated spectral sequence converges in the strong (i.e.,finite) sense to the cohomology HL n ( L | Q , M ) of the quotient cochain complexCL n ( L , M )[ − /CL n ( Q , M )[ − E p,q = Hom F ( Q p ⊗ L q +1 , M ) / Hom F ( Q p +1 ⊗ L q , M ) ∼ = Hom F ( Q p ⊗ I ⊗ L q , M ) , where the isomorphism is induced by the inclusion I ֒ → L . Since Hom F and ⊗ areadjoint functors, we obtain that E p,q = Hom F ( Q p ⊗ I , Hom F ( L q , M )) = Hom F ( Q p ⊗ I , CL q ( L , M )) . Next, we will determine the differential on E p,q : Lemma 3.3. The differential d on E p,q ∼ = Hom F ( Q p ⊗ I , CL q ( L , M )) is inducedby the coboundary operator d • on CL q ( L , M ) . More precisely, we have that d p,q ( f ) := d q CL q ( L ,M ) ◦ f for every linear transformation f ∈ Hom F ( Q p ⊗ I , CL q ( L , M ))) .Proof. The differentiald : F p CL n ( L , M )[ − / F p +1 CL n ( L , M )[ − →→ F p CL n +1 ( L , M )[ − / F p +1 CL n +1 ( L , M )[ − • . Thus, we have to examine which terms d ij ( c ), δ i ( c ), and ∂ ( c )composing the value d ( c ) are non-zero when we put an element of I within thefirst p + 1 entries.It is clear that d ij ( c ) = 0 for i, j ≤ p + 1 since this is true when the element of I is not involved in the product, as the number of elements is diminished by one,and it is also true when the element of I is in the product because I is an ideal.We then have d ij ( c ) = 0 for i ≤ p + 1 and j ≥ p + 2, since in case the element of I acts in the product, it acts trivially because of I ⊆ C ℓ ( L ). Furthermore, the terms δ i ( c ) for i ≤ p + 1 are zero since elements of I act trivially on M .Therefore, we are left with the terms composing the differential d | CL q ( L ,M ) . (cid:3) Consequently, the first term of the spectral sequence is E p,q = Hom F ( Q p ⊗ I , HL q ( L , M )) . Now we proceed to determine the differential d on E p,q . It is again induced bythe Leibniz coboundary operator. As before, classes in HL q ( L , M ) are representedby cocycles and thus the part of the Leibniz differential constituting the Leibnizcoboundary operator d q CL q ( L ,M ) is zero. The remaining terms constitute the dif-ferential on Q p ⊗ I , again since by the Cartan relations for Leibniz cohomology(see [26, Proposition 3.1] for the case of right Leibniz algebras and [10, Proposi-tion 1.3.2] for the case of left Leibniz algebras) a Leibniz algebra acts trivially onits cohomology. But one needs to be careful since the Cartan relations do only hold for q ≥ 1. Therefore, for an arbitrary bimodule M , Q will act non-triviallyon HL ( L , M ). On the other hand, if the bimodule M is symmetric, however, theaction is indeed trivial on HL ( L , M ).Note that in the proof of the preceding lemma, in all action terms on I oron Hom F ( I , HL ( L , M )) the action is from the left, thus, in order to interpretthe remaining terms as the Leibniz boundary operator with values in I , we haveto switch around the last action term. This is the reason for viewing I andHom F ( I , HL ( L , M ))) here as a symmetric Q -bimodule.Finally, we obtain from the Universal Coefficient Theorem (for example, see [35,Theorem 3.6.5]) that for q > E p,q = Hom F (HL p ( Q , I s ) , HL q ( L , M )) . Furthermore, for a finite-dimensional Leibniz algebra L and a symmetric Q -bimodule M , for all p, q ≥ 0, the E -term simplifies to E p,q = CL p ( Q , I ∗ ) ⊗ HL q ( L , M ) , and in this special case, for all p, q ≥ 0, the E -term is E p,q = HL p ( Q , ( I ) ∗ s ) ⊗ HL q ( L , M ) . This discussion proves the following results: Theorem 3.4. Let → I → L π → Q → be a short exact sequence of left Leibnizalgebras such that I ⊆ C ℓ ( L ) . Then I is in a natural way an anti-symmetric Q -bimodule via a · y := π − ( a ) y and y · a := yπ − ( a ) for every element a ∈ Q and everyelement y ∈ I . Moreover, there is a spectral sequence converging to HL • ( L | Q , M ) with second term E p,q = (cid:26) HL p ( Q , Hom F ( I , HL ( L , M )) s ) if p ≥ , q = 0Hom F (HL p ( Q , I s ) , HL q ( L , M )) if p ≥ , q ≥ for every Q -bimodule M . Corollary 3.5. If in Theorem 3.4 the Leibniz algebra L is finite dimensional andthe Q -bimodule M is symmetric, then, for any integers p, q ≥ , the E -term of thespectral sequence simply reads E p,q = HL p ( Q , ( I ∗ ) s ) ⊗ HL q ( L , M ) , where the linear dual I ∗ of I is a left L -module via ( x · f )( y ) := − f ( xy ) for everylinear form f ∈ I ∗ and any elements x ∈ L , y ∈ I . Remarks. (a) According to [14, Proposition 2.13], Theorem 3.4 applies to I := Leib( L )and Q := L Lie (see [31, Remark 4.2] for the analogous statement for Leibnizhomology). Note that in the cohomology space HL p ( Q , ( I ∗ ) s ), the left Q -module I ∗ is viewed as a symmetric bimodule while naturally it is ananti-symmetric Q -bimodule.(b) The higher differentials in the spectral sequence are again induced by theLeibniz coboundary operator d • . Observe that the spectral sequence ofCorollary 3.5 is isomorphic to the spectral sequence of the cochain bicom-plex CL • ( Q , ( I ∗ ) s ) ⊗ CL • ( L , M ). Therefore the description of the higher dif-ferentials can be adapted from [21] (see, in particular, Remark 3.2 therein).For example, it is clear, if one of the two differentials in the bicomplex is EIBNIZ COHOMOLOGY 25 zero, then all higher differentials vanish. We will see an instance of thiscase in Example C below.(c) One might wonder what one gets when one uses the filtration by the last p arguments instead of the first p arguments. It turns out that this spectralsequence has an E -term that is more difficult to describe (and which westated erroneously in a first version of this article), because one takes inthe E -term the cohomology of a complex which appears as coefficients inthe Leibniz cohomology that constitutes the E -term.As in the previous section, we illustrate the use of the spectral sequence ofTheorem 3.4 and the associated long exact sequence (see Proposition 3.1) by twoexamples.In the first example we compute the Leibniz cohomology of the smallest nilpotentnon-Lie left Leibniz algebra with trivial coefficients. Example C. Let F denote an arbitrary field, and let N := F e ⊕ F f be the two-dimensional nilpotent left (and right) Leibniz algebra over F with multiplicationdetermined by f f = e . Then Leib( N ) = F e , and thus N Lie is a one-dimensionalabelian Lie algebra. Hence HL n ( N Lie , F ) ∼ = F for every non-negative integer n .Moreover, we have that dim F HL ( N , F ) = 1 and dim F HL ( N , F ) = 1.Next, we compute the higher cohomology with the help of the spectral sequenceof Corollary 3.5. As observed in Remark (b) after Corollary 3.5, all higher differen-tials are zero in our case since the Leibniz coboundary operator of the abelianLie algebra with values in the trivial module vanishes. With the input datadim F HL ( N , F ) = 1 and dim F HL ( N , F ) = 1, we therefore get from the spectralsequence dim F HL ( N | N Lie , F ) = 1 and dim F HL ( N | N Lie , F ) = 2 . In order to be able to apply now the long exact sequence from Proposition 3.1and deduce the dimensions of the cohomology spaces, we want to argue that thissequence is split. In fact, it is split because the connecting homomorphism issurjective. This comes from the fact that the cochain complex CL • ( N Lie , F ) is one-dimensional in each degree and a generator can be hit via the connecting homomor-phism which is easy to see directly (take a cochain in CL n ( N | N Lie , F ) representedby an element in CL n +1 ( N , F ) with exactly one slot in e ∗ at the first place: theLeibniz product in this slot gives the only non-zero contribution). Consequently,the long exact sequence from Proposition 3.1 splits into short exact sequences0 → HL n ( N , F ) → HL n − ( N | N Lie , F ) → HL n +1 ( N Lie , F ) → , starting from n = 2, where the right-hand term is one-dimensional. These shortexact sequences, together with the spectral sequence where every differential is zero,permit us to determine all relative and absolute cohomology spaces. For example,we have dim F HL ( N , F ) = 1, and then dim F HL ( N | N Lie , F ) = 3, dim F HL ( N , F ) =2, and then dim F HL ( N | N Lie , F ) = 5, and so on. In general, we obtain by inductionthat dim F HL n ( N , F ) = 2 n − for every integer n ≥ F HL n ( N | N Lie , F ) =2 n − + 1 for every integer n ≥ algebras in Example A the Leibniz algebra in Example D is the hemi-semidirectproduct of two one-dimensional Lie algebras. It turns out that this somewhatsimplifies matters. Example D. Let F denote an arbitrary field, and let A := F h ⊕ F e be the two-dimensional supersolvable left Leibniz algebra over F with multiplication deter-mined by he = e . For any scalar λ ∈ F one can define a one-dimensional left A -module F λ := F λ with A -action defined by h · λ := λ λ and e · λ := 0. Notethat Leib( A ) = F e , and thus A Lie is a one-dimensional abelian Lie algebra. Thenwe obtain from [3, Lemma 1] and Theorem 2.6 thatdim F HL n ( A Lie , ( F λ ) s ) = (cid:26) λ = 0 and n is arbitrary0 otherwise . Moreover, we deduce from Lemma 1.4 (b) thatHL n ( A Lie , ( F λ ) a ) ∼ = HL n − ( A Lie , Hom F ( A Lie , F λ ) s ) ∼ = HL n − ( A Lie , ( F λ ) s )for every integer n ≥ 1, and thereforedim F HL n ( A Lie , ( F λ ) a ) = (cid:26) λ = 0 and n is arbitrary or if λ = 0 and n = 00 otherwise . In order to be able to apply the spectral sequence of Theorem 3.4, we firstcompute HL • ( A Lie , [Leib( A ) ∗ ] s ). Observe that the module Leib( A ) ∗ = F e ∗ ∼ = F − isnon-trivial irreducible, and furthermore it is viewed as a symmetric A Lie -bimodule.Hence from the above it follows that HL n ( A Lie , [Leib( A ) ∗ ] s ) = 0 for every non-negative integer n . This implies in turn that the spectral sequence of Theorem 3.4collapses at the E -term and thatHL n ( A | A Lie , ( F λ ) a ) = HL n ( A Lie , Hom F (Leib( A ) , HL ( A , ( F λ ) a )) s )= HL n ( A Lie , Hom F (Leib( A ) , F λ ) s )for all non-negative integers n , while HL n ( A | A Lie , ( F λ ) s ) = 0 for all n ≥ A -bimodule Hom F (Leib( A ) , F λ ) s ∼ = [ F λ − ] s . Wehave already observed in Example C that the long exact sequence of Proposition 3.1splits, and therefore we conclude from Proposition 3.1 thatHL n ( A , ( F λ ) s ) ∼ = HL n ( A Lie , ( F λ ) s )and HL n ( A , ( F λ ) a ) ∼ = HL n ( A Lie , ( F λ ) a ) ⊕ HL n ( A Lie , ( F λ − ) s )for all λ and all non-negative integers n . Consequently, we obtain thatdim F HL n ( A , ( F λ ) s ) = (cid:26) λ = 0 and n is arbitrary0 otherwise , anddim F HL n ( A , ( F λ ) a ) = (cid:26) λ = 0 , n is arbitrary or if λ = 0 , n = 00 otherwise . Remark. In particular, we have that dim F HL n ( A , F ) = 1 for every non-negativeinteger n . Note that this follows as well from the scheme of proof of Proposition 4.3in [31] by using the isomorphism between Leibniz homology and cohomology with EIBNIZ COHOMOLOGY 27 trivial coefficients. Indeed, the characteristic element ch( A ) ∈ HL ( A Lie , Leib( A ))of A is zero as Leib( A ) = F e ∼ = ( F ) a and HL ( A Lie , ( F ) a ) = 0. Since alsoHL • ( A Lie , [Leib( A ) ∗ ] s ) is zero, we can reason in the same way as Pirashvili does.4. Cohomology of semi-simple Leibniz algebras Recall that a left Leibniz algebra L is called semi-simple if Leib( L ) containsevery solvable ideal of L (see [14, Section 7]). In particular, a finite-dimensional leftLeibniz algebra L is semi-simple if, and only if, Leib( L ) = Rad( L ), where Rad( L )denotes the largest solvable ideal of L (see [14, Proposition 7.4]). Moreover, aleft Leibniz algebra L is semi-simple if, and only if, the canonical Lie algebra L Lie associated to L is semi-simple (see [14, Proposition 7.8]). Lemma 4.1. Let L be a finite-dimensional semi-simple left Leibniz algebra overa field of characteristic zero. Then [Leib( L ) ∗ ] L Lie s = 0 , where Leib( L ) ∗ is a left L -module, and thus a left L Lie -module, via ( x · f )( y ) := − f ( xy ) for every linearform f ∈ Leib( L ) ∗ and any elements x, y ∈ L .Proof. It follows from Levi’s theorem for Leibniz algebras (see [31, Proposition 2.4]or [4, Theorem 1]) that there exists a semi-simple Lie subalgebra s of L such that L = s ⊕ Leib( L ) (see also [15, Corollary 2.14]). Note that then L Lie ∼ = s . Since s is a Lie algebra and Leib( L ) ⊆ C ℓ ( L ), we obtain that ( s + x )( s + x ) = sx for anyelements s ∈ s and x ∈ Leib( L ). This shows that Leib( L ) = s Leib( L ). Now let ϕ ∈ [Leib( L ) ∗ ] s s be arbitrary. Since ( s · ϕ )( x ) = − ϕ ( sx ) for any ϕ ∈ Leib( L ) ∗ , s ∈ s ,and x ∈ Leib( L ), we conclude that ϕ [Leib( L )] = ϕ [ s Leib( L )] = 0, which proves theassertion. (cid:3) The first main result in this section is the Leibniz analogue of Whitehead’s van-ishing theorem for the Chevalley-Eilenberg cohomology of finite-dimensional semi-simple Lie algebras over a field of characteristic zero (see [9, Theorem 24.1] or [19,Theorem 10]). Note that in the special case of a Lie algebra, Theorem 4.2 is an im-mediate consequence of Whitehead’s classical vanishing theorem and Theorem 2.6. Theorem 4.2. Let L be a finite-dimensional semi-simple left Leibniz algebra overa field of characteristic zero. If M is a finite-dimensional L -bimodule such that M L = 0 , then HL n ( L , M ) = 0 for every non-negative integer n .Proof. According to Lemma 1.1, the hypothesis M L = 0 implies that M is symmet-ric. We can therefore use the spectral sequence of Corollary 3.5 with I := Leib( L )and Q := L Lie . The E -term reads E p,q = HL p ( Q , ( I ∗ ) s ) ⊗ HL q ( L , M ) . It follows from [14, Proposition 7.8] and the Ntolo-Pirashvili vanishing theorem forthe Leibniz cohomology of a finite-dimensional semi-simple Lie algebra over a fieldof characteristic zero (see [30, Th´eor`eme 2.6] and the sentence after the proof ofLemma 2.2 in [31]) that HL p ( Q , ( I ∗ ) s ) = 0 for every positive integer p . Hence thespectral sequence collapses, and we deduceHL n ( L | Q , M ) = ( I ∗ ) Q s ⊗ HL n ( L , M ) . By virtue of Lemma 4.1, the relative cohomology HL n ( L | Q , M ) vanishes for everynon-negative integer n , and thus we obtain from Proposition 3.1 in conjunction with [14, Proposition 4.1] and the Ntolo-Pirashvili vanishing theorem that HL n ( L , M ) ∼ =HL n ( Q , M ) = 0 for every non-negative integer n . (cid:3) Remark. It is possible to prove Theorem 4.2 without using the Ntolo-Pirashvilivanishing theorem. Namely, the first time the Ntolo-Pirashvili vanishing theorem isused in the above proof, one can instead use Lemma 4.1, Whitehead’s classical van-ishing theorem, and Theorem 2.6, and the second time, by hypothesis, it is enoughto apply just the last two results. As a consequence, the proof of Theorem 4.3 givesalso a new proof of the Ntolo-Pirashvili vanishing theorem.Next, we generalize the Ntolo-Pirashvili vanishing theorem from Lie algebras toarbitrary Leibniz algebras. The main tools in the proof are Corollary 1.3, Theo-rem 4.2, Corollary 1.5, and Lemma 1.4, where the second result and its use in thisproof seems to be new. Theorem 4.3. Let L be a finite-dimensional semi-simple left Leibniz algebra overa field of characteristic zero, and let M be a finite-dimensional L -bimodule. Then HL n ( L , M ) = 0 for every integer n ≥ , and there is a five-term exact sequence → M → HL ( L , M ) → M L Lie sym → Hom L ( L ad ,ℓ , M ) → HL ( L , M ) → . Moreover, if M is symmetric, then HL n ( L , M ) = 0 for every integer n ≥ .Proof. The proof is divided into three steps. First, we will prove the assertionfor symmetric L -bimodules. So suppose that M is symmetric. Since M is finite-dimensional, it has a composition series. It is clear that sub-bimodules and ho-momorphic images of a symmetric bimodule are again symmetric. By using thelong exact cohomology sequence, it is therefore enough to prove the second partof the theorem for finite-dimensional irreducible symmetric L -bimodules. So sup-pose now in addition that M is irreducible and non-trivial. Then we obtain fromCorollary 1.3 that M L = 0, and thus Theorem 4.2 yields that HL n ( L , M ) = 0for every non-negative integer n . Finally, suppose that M = F is the trivial irre-ducible L -bimodule. In this case it follows from Corollary 1.5 that HL n ( L , F ) ∼ =HL n − ( L , ( L ∗ ) s ) for every integer n ≥ 1. Since L Lie is perfect, we obtain fromCorollary 1.5 that( L ∗ ) L s ∼ = HL ( L , ( L ∗ ) s ) ∼ = HL ( L , F ) ∼ = H ( L Lie , F ) = 0 . Therefore another application of Theorem 4.2 yields thatHL n ( L , F ) ∼ = HL n − ( L , ( L ∗ ) s ) = 0for every integer n ≥ 1. This finishes the proof for symmetric L -bimodules.If M is anti-symmetric, then we obtain the assertion from Lemma 1.4 (b) andthe statement for symmetric bimodules. Finally, if M is arbitrary, then in theshort exact sequence 0 → M → M → M sym → n ( L , M ) = 0 for every integer n ≥ 2. Now we deducethe five-term exact sequence from the long exact cohomology sequence togetherwith [14, Corollary 4.2], [14, Corollary 4.4 (b)], and the symmetric case. (cid:3) Note that Theorem 4.3 contains [14, Theorem 7.15] as the special case n = 1and the second Whitehead lemma for Leibniz algebras as the special case n = 2. EIBNIZ COHOMOLOGY 29 But contrary to Chevalley-Eilenberg cohomology, Leibniz cohomology vanishes inany degree n ≥ Example E. Let g := sl ( F ) be the three-dimensional simple Lie algebra of traceless2 × F of characteristic p > 2. Moreover, let F p denote thefield with p elements, and let L ( n ) ( n ∈ F p ) denote the irreducible restricted g -module of heighest weight n . (If the ground field F is algebraically closed, thesemodules represent all isomorphism classes of restricted irreducible g -modules.) Itis well known (see [11, Theorem 4]) that H ( g , L ( p − ∼ = F ∼ = H ( g , L ( p − • ( g , M ) = 0 for every non-restrictedirreducible g -module. In fact, L ( p − 2) is the only irreducible g -module M suchthat H ( g , M ) = 0 or H ( g , M ) = 0.)We obtain from Proposition 2.2 thatHL ( g , L ( p − s ) ∼ = H ( g , L ( p − ∼ = F = 0and 0 = F ∼ = H ( g , L ( p − ֒ → HL ( g , L ( p − s ) . In particular, this shows that the Ntolo-Pirashvili vanishing theorem (and thereforealso Theorem 4.3) is not true over fields of prime characteristic. Remark. By using more sophisticated tools one can also say something aboutthe Leibniz cohomology of anti-symmetric irreducible g -bimodules, where again g := sl ( F ). We obtain from Lemma 1.4 (b) thatHL ( g , L ( n ) a ) ∼ = HL ( g , Hom F ( g , L ( n )) s ) ∼ = Hom F ( g , L ( n )) g and HL ( g , L ( n ) a ) ∼ = HL ( g , Hom F ( g , L ( n )) s ) ∼ = H ( g , Hom F ( g , L ( n ))) . Since g ∼ = L (2) is a self-dual g -module, we have the following isomorphisms of g -modules: Hom F ( g , L ( n )) ∼ = L (2) ⊗ L ( n ) . Let us first consider the case p > 3. Then we obtain from the modular Clebsch-Gordan rule (see [5, Theorem 1.11 (a)] or Satz a) in Chapter 5 of [17]) that L (2) ⊗ L (2) ∼ = L (4) ⊕ L (2) ⊕ L (0)and L (2) ⊗ L ( p − ∼ = (cid:26) L (3) ⊕ L (1) if p = 5 L ( p − ⊕ L ( p − ⊕ L ( p − 6) if p ≥ . Hence we conclude for p > ( g , L (2) a ) ∼ = ( L (2) ⊗ L (2)) g ∼ = L (0) g ∼ = F = 0and HL ( g , L ( p − a ) ∼ = H ( g , L (2) ⊗ L ( p − ∼ = H ( g , L ( p − ∼ = F = 0 . Let us now consider p = 3. Note that in this case L (2) is the Steinberg module ,i.e., L (2) is the unique projective irreducible restricted g -module. This implies that L (2) ⊗ L ( n ) is also projective for every highest weight n ∈ F . Then we obtain from the modular Clebsch-Gordan rule (cf. [5, Theorem 1.11 (b) and (c)] or Satz b) andc) in Chapter 5 of [17]) for p = 3 that L (2) ⊗ L ( n ) ∼ = L (2) if n ≡ P (1) if n ≡ ,P (0) ⊕ L (2) if n ≡ P ( n ) denotes the projective cover (and at the same time also the injectivehull) of L ( n ). As a consequence, we have that( L (2) ⊗ L ( n )) g ∼ = (cid:26) F if n ≡ n . Therefore, we obtain thatHL ( g , L (2) a ) ∼ = ( L (2) ⊗ L (2)) g ∼ = F = 0 . Moreover, by using the six-tem exact sequence relating Hochschild’s cohomologyof a restricted Lie algebra to its Chevalley-Eilenberg cohomology (see [18, p. 575]),we also conclude thatHL ( g , L (2) a ) ∼ = H ( g , L (2) ⊗ L (2)) ∼ = g ∗ ∼ = F = 0 . The next example shows that the Ntolo-Pirashvili vanishing theorem (and there-fore also Theorem 4.3) does not hold for infinite-dimensional modules. Example F. Let g := sl ( C ) be the three-dimensional simple complex Lie algebra oftraceless 2 × V ( λ ) ( λ ∈ C ) denote the Verma module of highestweight λ . (Here we identify every complex multiple of the unique fundamentalweight with its coefficient.) Verma modules are infinite-dimensional indecomposable g -modules (see, for example, [22, Theorem 20.2 (e)]). Furthermore, it is well known(see [22, Exercise 7 (b) & (c) in Section 7.2]) that V ( λ ) is irreducible if, and onlyif, λ is not a dominant integral weight (i.e., with our identification, λ is not anon-negative integer). Moreover, it follows from [36, Theorem 4.19] thatH n ( g , V ( λ )) ∼ = (cid:26) C if λ = − n = 1 , 20 otherwise . This in conjunction with Proposition 2.2 yields thatHL ( g , V ( − s ) ∼ = H ( g , V ( − ∼ = C = 0and 0 = C ∼ = H ( g , V ( − ֒ → HL ( g , V ( − s ) . In particular, the Ntolo-Pirashvili vanishing theorem (and therefore also Theo-rem 4.3) is not true for infinite-dimensional modules.We obtain as an immediate consequence of Theorem 4.3 the following general-ization of [14, Corollary 7.9]. Corollary 4.4. If L is a finite-dimensional semi-simple left Leibniz algebra over afield of characteristic zero, then HL n ( L , F ) = 0 for every integer n ≥ . Remark. It is well known that the analogue of Corollary 4.4 does not hold forthe Chevalley-Eilenberg cohomology of Lie algebras as H ( g , F ) = 0 for any finite-dimensional semi-simple Lie algebra g over a field F of characteristic zero (see [9,Theorem 21.1]). EIBNIZ COHOMOLOGY 31 Next, we apply Theorem 4.3 to compute the cohomology of a finite-dimensionalsemi-simple left Leibniz algebra over a field of characteristic zero with coefficientsin its adjoint bimodule and in its (anti-)symmetric counterparts. Theorem 4.5. For every finite-dimensional semi-simple left Leibniz algebra L overa field of characteristic zero the following statements hold: (a) HL n ( L , L s ) = (cid:26) Leib( L ) if n = 00 if n ≥ . (b) HL n ( L , L a ) = L if n = 0End L ( L ad ,ℓ ) if n = 1 , n ≥ where End L ( L ad ,ℓ ) denotes the vector space of endomorphisms of the leftadjoint L -module L ad ,ℓ . (c) HL n ( L , L ad ) = Leib( L ) if n = 0Hom L ( L ad ,ℓ , Leib( L )) if n = 1 , n ≥ where Hom L ( L ad ,ℓ , Leib( L )) denotes the vector space of homomorphismsfrom the left adjoint L -module L ad ,ℓ to the Leibniz kernel Leib( L ) consideredas a left L -module.Proof. (a): According to [14, Proposition 4.1] and [14, Proposition 7.5] we have thatthat HL ( L , L s ) = ( L s ) L = C ℓ ( L ) = Leib( L ). Moreover, we obtain the statementfor degree n ≥ ( L , L a ) = L , and it followsfrom [14, Corollary 4.4 (b)] that HL ( L , L a ) = End L ( L ad ,ℓ ). The remainder of theassertion is an immediate consequence of the first part of Theorem 4.3.(c): As for the symmetric adjoint bimodule, we obtain from [14, Proposition 4.1]and [14, Proposition 7.5] that HL ( L , L ad ) = ( L ad ) L = C ℓ ( L ) = Leib( L ). Next, byapplying the five-term exact sequence of Theorem 4.3 to the adjoint L -bimodule M := L ad , we deduce thatHL ( L , L ad ) ∼ = Hom L ( L ad ,ℓ , Leib( L )) , as the third term is L L Lie Lie = C ( L Lie ) = 0. Finally, the assertion for degree n ≥ (cid:3) Remark. Note that the vanishing part of Theorem 4.5 (c) confirms a generalizationof the conjecture at the end of [1]. Moreover, parts (a) and (b) of Theorem 4.5 showthat the statements in Theorem 4.3 are best possible.In particular, one can derive from Theorem 4.5 (c) that finite-dimensional semi-simple non-Lie Leibniz algebras over a field of characteristic zero have outer deriva-tions. In this respect non-Lie Leibniz algebras behave differently than Lie algebras(see, for example, [22, Theorem 5.3]). Corollary 4.6. Every finite-dimensional semi-simple non-Lie Leibniz algebra overa field of characteristic zero has derivations that are not inner. Proof. If one applies the contravariant functor Hom F ( − , Leib( L )) to the short exactsequence 0 → Leib( L ) → L ad → L Lie → L -modules, one obtains the short exactsequence0 → Hom F ( L Lie , Leib( L )) → Hom F ( L ad ,ℓ , Leib( L )) → Hom F (Leib( L ) , Leib( L )) → L -modules. Then the long exact cohomology sequence in conjunction withLemma 1.4 (a) yields the long exact sequence0 → Hom L ( L Lie , Leib( L )) → Hom L ( L ad ,ℓ , Leib( L )) → Hom L (Leib( L ) , Leib( L )) → f HL ( L , Hom F ( L Lie , Leib( L ))) = HL ( L , Hom F ( L Lie , Leib( L )) s ) . According to the second part of Theorem 4.3, the fourth term is zero. Since thethird term contains the identity map, it is non-zero as by hypothesis L is a not aLie algebra. Hence in this case the second term is non-zero, and we obtain fromTheorem 4.5 (c) that HL ( L , L ad ) ∼ = Hom L ( L ad ,ℓ , Leib( L )) = 0. (cid:3) Remark. After the submission of our paper we became aware of the preprint [6] inwhich the authors introduce a more general concept of inner derivations for Leibnizalgebras than in our paper. Namely, a derivation D of a left Leibniz algebra L iscalled inner if there exists an element x ∈ L such that Im( D − L x ) ⊆ Leib( L ). Thenit is shown that every derivation of a finite-dimensional semi-simple Lie algebra overa field of characteristic zero is inner in this more general sense (see [6, Theorem 3.3]).In the same way as at the end of Section 2 for the infinite-dimensional two-sidedWitt algebra, by using [2, Th´eor`eme 3] in conjunction with Theorem 4.5 (c), oneobtains the rigidity of any finite-dimensional semi-simple Lie algebra as a Leibnizalgebra. Corollary 4.7. Every finite-dimensional semi-simple left Leibniz algebra over analgebraically closed field of characteristic zero is rigid as a Leibniz algebra. Acknowledgments. Most of this paper was written during a sabbatical leave ofthe first author in the Fall semester 2018. He is very grateful to the University ofSouth Alabama for giving him this opportunity.The first author would also like to thank the Laboratoire de math´ematiques JeanLeray at the Universit´e de Nantes for the hospitality and the financial support inthe framework of the program D´efiMaths during his visit in August and September2018. Moreover, the first author wishes to thank Henning Krause and the BIREPgroup at Bielefeld University for the hospitality and the financial support during hisvisit in October and November 2018 when large portions of the paper were written.Both authors would like to thank Bakhrom Omirov for useful discussions. We arealso grateful to Teimuraz Pirashvili for spotting a mistake in a previous version ofthe manuscript and to Geoffrey Powell for his help in understanding this mistake aswell as for several useful remarks that improved the present manuscript. Finally, wewould like to thank an anonymous referee for correcting a mistake in the submittedversion. 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Tolpygo: Cohomologies of parabolic Lie algebras (in Russian), Mat. Zametki (1972),251–255, English transl. Math. Notes (1972), 585–587.[35] C. A. Weibel: An introduction to homological algebra , Cambridge Studies in Advanced Math-ematics, vol. , Cambridge University Press, Cambridge, 1994.[36] F. L. Williams: The cohomology of semisimple Lie algebras with coefficients in a Vermamodule, Trans. Amer. Math. Soc. (1978), 115–127. Department of Mathematics and Statistics, University of South Alabama, Mobile,AL 36688-0002, USA Email address : [email protected] Laboratoire de math´ematiques Jean Leray, UMR 6629 du CNRS, Universit´e de Nantes,2, rue de la Houssini`ere, F-44322 Nantes Cedex 3, France Email address ::