Ext groups in the category of bimodules over a simple Leibniz algebra
aa r X i v : . [ m a t h . A T ] J un EXT GROUPS IN THE CATEGORY OF BIMODULES OVER A SIMPLE LEIBNIZALGEBRA
JEAN MUGNIERY AND FRIEDRICH WAGEMANN [2010]Primary 17A32; Secondary 17B56Leibniz cohomology, Chevalley-Eilenberg cohomology, change of rings spectral sequence, cohomologyvanishing, semi-simple Leibniz algebra, Whitehead theorem, Ext category of Leibniz bimodules
Abstract.
In this article, we generalize Loday and Pirashvili’s [10] computation of the
Ext -category ofLeibniz bimodules for a simple Lie algebra to the case of a simple (non Lie) Leibniz algebra. Most ofthe arguments generalize easily, while the main new ingredient is the Feldvoss-Wagemann’s cohomologyvanishing theorem for semi-simple Leibniz algebras.
Introduction
The goal of this article is to present a new result in the theory of representations of Leibniz algebras,namely to compute Ext groups between finite dimensional simple bimodules of a simple (non Lie) Leibnizalgebra over a field of characteristic . Leibniz algebras are a generalization of Lie algebras, discovered byA. Bloh in the 1960s, where one does not require the bracket to be antisymmetric. They were rediscoveredby J.-L. Loday [8] in the 1990s when, while trying to lift the boundary operator of Chevalley-Eilenberghomology from d : Λ n g −→ Λ n − g to ˜ d : g ⊗ n −→ g ⊗ n − , he noticed that the only property needed to showthat ˜ d ◦ ˜ d = 0 was the Leibniz identity of the bracket, that is: [ x, [ y, z ]] = [[ x, y ] , z ] + [ y, [ x, z ]] ∀ x, y, z ∈ g (see [8] for a survey of the subject).J.-L. Loday and T. Pirashvili studied the Leibniz representations of semi-simple Lie algebras in [10],and established in that paper the following theorem (Theorem 3.1): Theorem -1.1.
Let g be a finite dimensional simple Lie algebra over a field of characteristic , and let L ( g ) denote the category of finite dimensional Leibniz representations of g . The simple objects in L ( g ) are exactly the representations of the form M a and N s , where M , and N are simple right g -modules.All groups Ext UL ( g ) ( M, N ) between simple finite dimensional representations M , N are zero, except Ext UL ( g ) ( g s , g a ) which is one-dimensional. Moreover, Ext UL ( g ) ( M s , N a ) ≃ Hom U ( g ) ( M, ˆ N ) where ˆ N = Coker ( h : N −→ Hom ( g , N )) , h ( n )( x ) = [ n, x ] and all other groups Ext UL ( g ) ( M, N ) between simple finite dimensional representations M , N are zero.This theorem shows in particular that, contrary to the representations of semi-simple Lie algebras, thecategory of Leibniz bimodules of a semi-simple Lie algebra is not semi-simple.The aim of our article is to generalize this result to Leibniz representations of simple (non Lie) Leibnizalgebras by closely following [10], and making the adequate changes whenever necessary. The key in doingso is a theorem of J. Feldvoss and F. Wagemann, namely Theorem 4.2 of [4], assuring the vanishing ofLeibniz cohomology needed in the proof of our main theorem: Theorem -1.2.
Let h be a finite-dimensional semisimple left Leibniz algebra over a field of characteristiczero, and let M be a finite-dimensional h -module. Then HL n ( h , M ) = 0 for every integer n ≥ , and if M is symmetric, then HL n ( h , M ) = 0 for every integer n ≥ . Interestingly, this result represents a continuation of other vanishing theorems. First Whitehead’sTheorem giving the vanishing of Chevalley-Eilenberg cohomology of a semi-simple Lie algebra with valuesin a finite dimensional g -module whose invariants are trivial. Then P. Ntolo in [11] and T. Pirashvili in [12]independently proved results about Leibniz (co)homology of Lie algebras, while the authors of [4] provedthe vanishing of Leibniz cohomology of semi-simple Leibniz algebras.This allows us to prove the main theorem of the article, namely: Theorem -1.3.
Let h be a finite dimensional simple Leibniz algebra over a field of characteristic zero k . Allgroups Ext UL ( h ) ( M, N ) between simple finite dimensional h -bimodules are zero, except Ext UL ( h ) ( M s , N a ) ,with M ∈ { Leib ( h ) ⋆ , h ⋆Lie } and N ∈ { Leib ( h ) , h Lie } which is one dimensional.Moreover, we have that: • Ext UL ( h ) ( M s , k ) , and Ext UL ( h ) ( k, N a ) are one dimensional, for M and N ∈ { Leib ( h ) , h Lie } ; • Ext UL ( h ) ( M s , N a ) ≃ Hom U ( h Lie ) ( M, b N ) , where b N := Coker ( h : N −→ Hom ( h , N )) h ( n )( x ) := [ n, x ] R • All other groups
Ext UL ( h ) ( M, N ) between simple finite-dimensional h -bimodules M and N arezero.Here Leib ( h ) denotes the two-sided ideal generated by the elements [ x, x ] for x ∈ h . We see that, whenwe do not restrict ourselves to Lie algebras, there are more non-trivial Ext groups. Moreover, as a corollaryof this theorem one can show that the Ext dimension of the category of finite dimensional bimodules overa semi-simple Leibniz algebra is again . Acknowledgements:
The authors would like to thank A. Djament for pointing out a simpler way to show that the Ext inthe category of finite dimensional bimodules over a simple Leibniz algebra are well defined.1.
Leibniz Algebras
In this section, we introduce the objects in which we are interested, as well as some of their basicproperties. All of this material is due to Loday and Pirashvili. For more results on Leibniz algebras asnon-associative algebras see [3], and see [9] for results about their (co)homology.
Definition 1.1.
A (left)
Leibniz algebra over a field k is a vector space h equipped with a bilinear map: [ − , − ] : h × h −→ h called Leibniz bracket, that satisfies the (left) Leibniz identity: (1) [ x, [ y, z ]] = [[ x, y ] , z ] + [ y, [ x, z ]] ∀ x, y, z ∈ h With this definition, we see that Leibniz algebras are indeed a generalization of Lie algebras, as it is notdifficult to check that if we impose the anticommutativity of the bracket, the Jacobi and Leibniz identitiesare equivalent.
Remark 1.2.
We can also define a right Leibniz algebra by asking our bracket to satisfy the right Leibnizidentity instead: [[ x, y ] , z ] = [[ x, z ] , y ] + [ x, [ y, z ]] , but we will only be concerned with left Leibniz algebras.For every Leibniz algebra h , we have short exact sequence:(2) −→ Leib ( h ) −→ h −→ h Lie −→ where Leib ( h ) is the Leibniz kernel of h , that is the two-sided ideal generated by the elements [ x, x ] for x ∈ h ; and h Lie := h / Leib ( h ) . h Lie is a Lie algebra, called the canonical Lie algebra associated to h Definition 1.3.
A left Leibniz algebra is called simple if 0,
Leib ( h ) , and h are the only two sided ideals of h , and Leib ( h ) $ [ h , h ] . XT GROUPS IN THE CATEGORY OF BIMODULES OVER A SIMPLE LEIBNIZ ALGEBRA 3
This is not the only definition of simplicity: one can also only require that and h are the only ideals of h , but with this definition, all simple Leibniz algebras are in fact Lie algebras, see the beginning of section7 of [3], and the references therein. We will also need the Proposition 7.2 of [3], namely: Proposition 1.4. If h is a simple Leibniz algebra, then h Lie is a simple Lie algebra and
Leib ( h ) is a simple h Lie -module.We now give the definition of the notion of Leibniz modules and bimodules.
Definition 1.5.
Let h be a Leibniz algebra. A h -bimodule is a vector space M over k equipped with twobilinear maps: [ − , − ] L : h × M −→ M and [ − , − ] R : M × h −→ M which satisfy the following relations ∀ x, y ∈ h , ∀ m ∈ M : (LLM) [ x, [ y, m ] L ] L = [[ x, y ] , m ] L + [ y, [ x, m ] L ] L (LML) [ x, [ m, y ] R ] L = [[ x, m ] L , y ] R + [ m, [ x, y ]] R (MLL) [ m, [ x, y ]] R = [[ m, x ] R , y ] R + [ x, [ m, y ] R ] L We define a left h -module as being a vector space M over k equipped with a bilinear map: [ − , − ] L : h × M −→ M satisfying the relation (LLM) of Definition 1.5. Definition 1.6.
Let h be a Leibniz algebra, and M a Leibniz bimodule.If [ x, m ] L = − [ m, x ] R ∀ x ∈ h , ∀ m ∈ M then M is said to be symmetric and denoted M s .If [ m, x ] R = 0 ∀ x ∈ h , ∀ m ∈ M then M is said to be antisymmetric and denoted M a .If M is both symmetric and antisymmetric, then M is trivial . For every h -bimodule M , there is a short exact sequence of h -bimodules:(3) −→ M −→ M −→ M/M −→ where M = Span k ([ x, m ] L + [ m, x ] R ) , see (1.10) of [9]. Note that by construction M/M is a symmetric h -bimodule, and that M is an antisymmetric h -bimodule. Moreover, if we consider h as a h -bimoduleusing the adjoint action, then the short exact sequences (2) and (3) coincide.If M is a h -bimodule (in fact this works even when M is only a left h -module), then it has a natural h Lie -module structure (in the Lie sense). Indeed one can define a left action of h Lie as follows: h Lie × M −→ M (¯ x, m ) [ x, m ] L Conversely, if M is a h Lie -module, there are two natural ways to see it as a h -bimodule. We first see itas a left h -module via the projection h −→ h Lie , and then we impose our right action to be either trivial,or to be the opposite of the left action, yielding respectively an antisymmetric bimodule, or a symmetricone. Knowing this we can state the following Theorem:
Theorem 1.7.
The simple objects in the category of h -bimodules of finite dimension are exactly themodules of the form M a and M s , where M is a finite dimensional simple h Lie -module.
JEAN MUGNIERY AND FRIEDRICH WAGEMANN
The proof follows easily from the existence of the short exact sequence (3), and the fact that
Leib ( h ) acts trivially from the left (i.e. is contained in the left center).We now introduce the notion of the universal enveloping algebra of a Leibniz algebra, see (2.1) of [9](but note that the authors work with right Leibniz algebras).
Definition 1.8.
Let h be a Leibniz algebra. Given two copies h l and h r of h generated respectively bythe elements l x and r x for x ∈ h , we define the universal enveloping algebra of h as the unital associativealgebra: U L ( h ) := T ( h l ⊕ h r ) / I where T ( h l ⊕ h r ) := ∞ M n =0 ( h l ⊕ h r ) ⊗ n is the tensor algebra of h l ⊕ h r and I is the two-sided ideal of h generatedby the elements : l [ x,y ] − l x ⊗ l y + l y ⊗ l x r [ x,y ] − l x ⊗ r y + r y ⊗ l x r y ⊗ ( l x + r x ) For a Lie algebra g , there is an equivalence between being a g -module and being a U ( g ) -module, where U ( g ) is the universal enveloping algebra of the Lie algebra g . The following theorem allows us to establishthe same kind of connection between the structure of h -bimodule and left U L ( h ) -module. Theorem 1.9.
Let h be a Leibniz algebra. There is an equivalence of categories between the category of h -bimodules and the category of U L ( h ) -modules.For a proof see Theorem (2.3) of [9] (note once again that the authors work with right Leibniz algebras).This actually tells us, given one of the two structures, how to obtain the other: the action of l x correspondsto the left action [ x, − ] L while the action r y corresponds to the right action [ − , y ] R .Another nice property of this universal enveloping algebra is that we can establish a connection between U ( h Lie ) -modules and U L ( h ) -modules. To this end we define the following algebras homomorphisms: d : U L ( h ) −→ U ( h Lie ) d ( l x ) = ¯ xd ( r x ) = 0 and: d : U L ( h ) −→ U ( h Lie ) d ( l x ) = ¯ xd ( r x ) = − ¯ x With these, given a U ( h Lie ) -module, we can see it as a U L ( h ) -module either via d or via d . The formergives an antisymmetric h -bimodule, while the latter gives a symmetric h -bimodule. Moreover, since theyare surjective (their image contains the generators of U ( h Lie ) ), this allows us to consider U ( h Lie ) as thequotient U L ( h ) /Ker ( d i ) for i ∈ { , } .Being a generalization of Lie algebras, Leibniz algebras are equiped with a generalization of Chevalley-Eilenberg cohomology, namely Leibniz cohomology which was discovered by Loday. Let h be a Leibnizalgebra, and M be a h -module. We give the (left version of the) definition of the cochain complex CL n ( h , M ) , dL n } n ≥ from [9] (1.8), namely: CL n ( h , M ) = Hom ( h ⊗ n , M ) dL n : CL n ( h , M ) −→ CL n +1 ( h , M ) XT GROUPS IN THE CATEGORY OF BIMODULES OVER A SIMPLE LEIBNIZ ALGEBRA 5 with: dL n ω ( x , ..., x n ) = n − X i =0 ( − i [ x i , ω ( x , ..., ˆ x i , ..., x n )] L + ( − n − [ ω ( x , ..., x n − ) , x n ] R + X ≤ i Let h be a Leibniz algebra, and M be a h -module. The cohomology of h with coefficientsin M is the cohomology of the cochain complex { CL n ( h , M ) , dL n } n ≥ . HL n ( h , M ) = H n ( { CL n ( h , M ) , dL n } n ≥ ) ∀ n ≥ Remark 1.11. By definition CL ( h , M ) = M and dL m ( x ) = − [ m, x, ] R . Therefore, we have: HL ( h , M ) = { m ∈ M, [ m, x ] R = 0 ∀ x ∈ h } This is the submodule of right invariants. Note that if M is antisymmetric, then HL ( h , M ) = M .2. Ext in the category of Leibniz bimodules We are now interested in computing the Ext groups in the category of h -bimodules. From now on, we willconsider a finite-dimensional left Leibniz algebra h over a field of characteristc zero k . The definition of themorphisms d and d , together with the change of rings spectral sequence constructed in the subsections1 to 4 of Chapter XVI from [2], yield the following spectral sequences: E pq = Ext pU ( h Lie ) (cid:16) Y, Ext qUL ( h ) ( U ( h Lie ) a , X ) (cid:17) = ⇒ Ext p + qUL ( h ) ( Y a , X ) (S1) E pq = Ext pU ( h Lie ) (cid:16) Z, Ext qUL ( h ) ( U ( h Lie ) s , X ) (cid:17) = ⇒ Ext p + qUL ( h ) ( Z s , X ) (S2)where X is an h -bimodule, and Y and Z are left h -modules.For a Lie algebra g , we have the following isomorphism Ext ∗ U ( g ) ( M, N ) ≃ H ∗ ( g , Hom ( M, N )) , which wecan use to rewrite (S1) and (S2) as: E pq = H p (cid:16) h Lie , Hom ( Y, Ext qUL ( h ) ( U ( h Lie ) a , X ) (cid:17) = ⇒ Ext p + qUL ( h ) ( Y a , X ) (S1) E pq = H p (cid:16) h Lie , Hom ( Z, Ext qUL ( h ) ( U ( h Lie ) s , X ) (cid:17) = ⇒ Ext p + qUL ( h ) ( Z s , X ) (S2)Moreover, by Theorem (3.4) of [9], we have an isomorphism Ext ∗ UL ( h ) ( U ( h Lie ) a , X ) ≃ HL ∗ ( h , X ) This gives us the proposition: Proposition 2.1. Let h be a Leibniz algebra, let X be a h -bimodule, and Y and Z be left h -modules.There are two spectral sequences: E pq = H p ( h Lie , Hom ( Y, HL q ( h , X ))) = ⇒ Ext p + qUL ( h ) ( Y a , X ) (S1) E pq = H p (cid:16) h Lie , Hom (cid:16) Z, Ext qUL ( h ) ( U ( h Lie ) s , X ) (cid:17)(cid:17) = ⇒ Ext p + qUL ( h ) ( Z s , X ) (S2)In the previous proposition, we were able to identify Ext ∗ UL ( h ) ( U ( h Lie ) a , X ) to the Leibniz cohomology HL ∗ ( h , X ) . What about Ext ∗ UL ( h ) ( U ( h Lie ) s , X ) ? The following result will give a generalization of Propo-sition 2.3 of [10], in order to give a relation between Ext ∗ UL ( h ) ( U ( h Lie ) s , X ) and Leibniz cohomology. Inorder to do so, we will have to introduce a shift in the homological degree which will be responsible fornontrivial Ext groups in what will follow. JEAN MUGNIERY AND FRIEDRICH WAGEMANN Proposition 2.2. Let h be a Leibniz algebra, and M be a h -bimodule. There are isomorphisms: Ext i +1 UL ( h ) ( U ( h Lie ) s , M ) ≃ Hom ( h , HL i ( h , M )) for i > 0 ≃ Coker ( f ) for i = 0 ≃ Ker ( f ) for i = -1where f : M −→ Hom ( h , HL ( h , M )) is given by: f ( m )( h ) = [ h, m ] L + [ m, h ] R ∀ h ∈ h , ∀ m ∈ M Proof. Let M be a h -bimodule, and f : M −→ Hom ( h , HL ( h , M )) f ( m )( h ) = [ h, m ] L + [ m, h ] R We first want to show that Ext UL ( h ) ( U ( h Lie ) s , M ) = Ker ( f ) . But by definition Ext UL ( h ) ( U ( h Lie ) s , M ) = Hom UL ( h ) ( U ( h Lie ) s , M ) We then define the map: ev : Hom UL ( h ) ( U ( h Lie ) s , M ) −→ Mϕ ϕ (1) which is an isomorphism onto Ker ( f ) , of inverse: µ : Ker ( f ) −→ Hom UL ( h ) ( U ( h Lie ) s , M ) m ϕ m : (1 m ) This gives the degree zero equality of the proposition.We now want to show that Ext UL ( h ) ( U ( h Lie ) s , M ) = Coker ( f ) .Consider U L ( h ) ⊗ h as a left U L ( h ) -module with the following action ∀ x ∈ h , ∀ r, s ∈ U L ( h ) : s. ( r ⊗ x ) = sr ⊗ x Define a homomorphism of left U L ( h ) -modules by: f : U L ( h ) ⊗ h −→ U L ( h )1 ⊗ h l h + r h Then f factors through f : U ( h Lie ) a ⊗ h −→ U L ( h ) . Indeed we have the following commutativediagram:(D) U L ( h ) ⊗ h U L ( h ) U ( h Lie ) a ⊗ h f d ⊗ id f and define f ( d ( x ) ⊗ h ) := f ( x ⊗ h ) which is well-defined: if x, y ∈ U L ( h ) are such that d ( x ) = d ( y ) ,then f ( x ⊗ h ) = f ( y ⊗ h ) . Indeed if x − y ∈ Ker ( d ) , then x = y + ¯ z with ¯ z ∈ h r z , z ∈ h i . Therefore, therelation r y ( l x + r x ) = 0 in U L ( h ) implies that f ( x ⊗ h ) = f ( y ⊗ h ) .We claim that f is injective. This follows from the diagram (D) and the fact that Ker ( f ) = Ker ( d ⊗ id ) .This therefore gives us the following short exact sequence: U ( h Lie ) a ⊗ h U L ( h ) Cokerf f XT GROUPS IN THE CATEGORY OF BIMODULES OVER A SIMPLE LEIBNIZ ALGEBRA 7 But by construction, Im ( f ) is the left ideal h l x + r x | x ∈ h i , which is equal to Ker ( d ) (see Section 1).This implies that Coker ( f ) is the quotient U L ( h ) /Ker ( d ) , that is Im ( d ) , and the short exact sequenceabove becomes: U ( h Lie ) a ⊗ h U L ( h ) U ( h Lie ) s f This short exact sequence yields the following long exact sequence in cohomology: −→ Hom UL ( h ) ( U ( h Lie ) s , M ) −→ Hom UL ( h ) ( U L ( h ) , M ) −→ Hom UL ( h ) ( U ( h Lie ) a ⊗ h , M ) −→ Ext UL ( h ) ( U ( h Lie ) s , M ) −→ Ext UL ( h ) ( U L ( h ) , M ) −→ Ext UL ( h ) ( U ( h Lie ) a ⊗ h , M ) −→ Ext UL ( h ) ( U ( h Lie ) s , M ) −→ . . . Now, by noticing the obvious identification Hom UL ( h ) ( U L ( h ) , M ) = M , and the fact that, U L ( h ) being afree U L ( h ) -module, it is projective and therefore Ext UL ( h ) ( U L ( h ) , M ) = 0 , we can extract the followingexact sequence: → Hom UL ( h ) ( U ( h Lie ) s , M ) → M → Hom UL ( h ) ( U ( h Lie ) a ⊗ h , M ) → Ext UL ( h ) ( U ( h Lie ) s , M ) → To obtain the desired isomorphism, we want to relate it to the exact sequence we get from f : −→ Ker ( f ) −→ M −→ Hom ( h , HL ( h , M )) −→ Coker ( f ) −→ and conclude by using the 5-lemma. We can send M onto M via the identity map. We then construct anisomorphism Hom UL ( h ) ( U ( h Lie ) a ⊗ h , M ) −→ Hom ( h , HL ( h , M )) Notice that since U ( h Lie ) a ⊗ h is a quotient of U L ( h ) ⊗ h , it is generated, as a U L ( h ) -module, by theelements ⊗ h , for h ∈ h . We can now define a map: Hom UL ( h ) ( U ( h Lie ) a ⊗ h , M ) −→ Hom ( h , HL ( h , M )) ϕ ˜ ϕ where ˜ ϕ ( h ) := ϕ (1 ⊗ h ) , for h ∈ h . The image of ˜ ϕ lies in HL ( h , M ) , for: (cid:2) ˜ ϕ ( h ) , h ′ (cid:3) R = (cid:2) ϕ (1 ⊗ h ) , h ′ (cid:3) R = ϕ ( r h ′ . (1 ⊗ h ))= 0 using the fact that ϕ is a U L ( h ) -morphism and the fact that we are considering the U L ( h ) -module U ( h Lie ) a ⊗ h .We can then construct its inverse, by: Hom ( h , HL ( h , M )) −→ Hom UL ( h ) ( U ( h Lie ) a ⊗ h , M ) u ϕ u with ϕ u : ¯ x ⊗ h x.u ( h ) where ¯ x denotes the class of x ∈ U L ( h ) in the quotient U ( h Lie ) a (see Section 1).This yields the following diagram: M Hom UL ( h ) ( U ( h Lie ) a ⊗ h , M ) Ext UL ( h ) ( U ( h Lie ) s , M ) 0 0 M Hom ( h , HL ( h , M )) Coker ( f ) 0 0 ∼ ( ∗ ) f where the arrow ( ∗ ) : Ext UL ( h ) ( U ( h Lie ) s , M ) −→ Coker ( f ) is given by functoriality of the Coker .To conclude, we just need to prove that this diagramm is commutative. It is sufficient to show that itis the case for the square: M Hom UL ( h ) ( U ( h Lie ) a ⊗ h , M ) M Hom ( h , HL ( h , M )) ∼ f JEAN MUGNIERY AND FRIEDRICH WAGEMANN Notice that for the arrow M −→ Hom UL ( h ) ( U ( h Lie ) a ⊗ h , M ) we identified M ≃ Hom UL ( h ) ( U L ( h ) , M ) via the map m ( ψ m : u u.m ) . This arrow is therefore given by ψ m ψ m ◦ f , that is: ¯ u ⊗ x ψ m ( f (¯ u ⊗ x )) = ψ m ( f ( d ( u ) ⊗ x ))= ψ m ( f ( u ⊗ x ))= ψ m ( u ( l x + r x ))= u ( l x + r x ) .m Since U ( h Lie ) a ⊗ h is generated as a U L ( h ) -module by the elements ⊗ x for x ∈ h , it is enough to checkthe commutativity of the diagram only on these elements. By explicitly writing the maps in question weget: m = ψ m (1 ⊗ x ( l x + r x ) .m ) m ( x [ x, m ] L + [ m, x ] R ) f which by Theorem 1.9 proves the commutativity of the square and therefore of the diagram. The 5-lemmathen tells us the arrow ( ∗ ) is an isomorphism, and we obtain the second isomorphism of the proposition.To get the higher degree isomorphisms, notice that the long exact sequence in cohomology we foundearlier goes as follow: . . . → Ext iUL ( h ) ( U L ( h ) , M ) −→ Ext iUL ( h ) ( U ( h Lie ) a ⊗ h , M ) −→ Ext i +1 UL ( h ) ( U ( h Lie ) , M ) −→ Ext i +1 UL ( h ) ( U L ( h ) , M ) → . . . But U L ( h ) being a free U L ( h ) -module, it is projective, hence Ext iUL ( h ) ( U L ( h ) , M ) = Ext i +1 UL ( h ) ( U L ( h ) , M )= 0 and this for all i . We thus obtain: −→ Ext iUL ( h ) ( U ( h Lie ) a ⊗ h , M ) −→ Ext i +1 UL ( h ) ( U ( h Lie ) , M ) −→ Now, in order to conclude, we use the fact that: Ext iUL ( h ) ( U ( h Lie ) a ⊗ h , M ) = Hom ( h , Ext iUL ( h ) ( U ( h Lie ) a , M )) which is obtained from the classical Hom/Tens adjunction, and the isomorphism given in Theorem (3.4)of [9]: Ext iUL ( h ) ( U ( h Lie ) a , M ) ≃ HL i ( h , M ) This gives us all the promised isomorphisms, therefore concluding the proof. (cid:3) We can now compute the Ext groups in the category of h -bimodules and we will see that nontrivial Ext groups appear whenever the degree shift from Proposition 2.2 is happening. Theorem 2.3. Let h be a finite dimensional simple Leibniz algebra over a field of characteristic zero k . Allgroups Ext UL ( h ) ( M, N ) between simple finite dimensional h -bimodules are zero, except Ext UL ( h ) ( M s , N a ) ,with M ∈ { Leib ( h ) ⋆ , h ⋆Lie } and N ∈ { Leib ( h ) , h Lie } , which is one dimensional.Moreover, we have that: • Ext UL ( h ) ( M s , k ) , and Ext UL ( h ) ( k, N a ) are one dimensional, for M and N ∈ { Leib ( h ) , h Lie } ; • Ext UL ( h ) ( M s , N a ) ≃ Hom U ( h Lie ) ( M, b N ) , where b N := Coker ( h : N −→ Hom ( h , N )) h ( n )( x ) := [ n, x ] R XT GROUPS IN THE CATEGORY OF BIMODULES OVER A SIMPLE LEIBNIZ ALGEBRA 9 • All other groups Ext UL ( h ) ( M, N ) between simple finite-dimensional h -bimodules M and N arezero. Proof. We will compute Ext ∗ UL ( h ) ( M, N ) for every combination of finite-dimensional h -bimodules M and N , and then use the knowledge of the Chevalley-Eilenberg cohomology for simple Lie algebras to conclude. • Case 1: M = N = k is the trivial h -bimodule.We apply Proposition 2.1 to Y = X = k . By Theorem -1.2, HL q ( h , k ) = 0 for q ≥ , since k beingtrivial, it is also symmetric. Therefore, we obtain: Ext ∗ UL ( h ) ( k, k ) ≃ H ∗ ( h Lie , k ) • Case 2: M = k is the trivial h -bimodule, and N is a nontrivial simple symmetric h -bimodule.We apply Proposition 2.1 to Y = k , and X = N . Once again by Theorem -1.2, we get: Ext nUL ( h ) ( k, N s ) = 0 for n ≥ • Case 3: M = k is the trivial h -bimodule, and N is a nontrivial simple antisymmetric h -bimodule.We have: HL q ( h , N a ) ≃ for q > 1, by Theorem -1.2 ≃ Hom U ( h Lie ) ( h , N ) for q = 1, by Lemma 1.5 of [4] ≃ N for q = 0, since N is antisymmetricSince N is a nontrivial simple antisymetric h -bimodule, it is also a nontrivial simple h Lie -module,and therefore H ∗ ( h Lie , N ) = 0 by Whitehead’s theorem. Now using Proposition 2.1, we find: Ext ∗ UL ( h ) ( k, N a ) ≃ H ∗− ( h Lie , Hom U ( h Lie ) ( h , N )) ≃ H ∗− ( h Lie , k ) ⊗ Hom U ( h Lie ) ( h , N ) The second isomorphism is given in [5], Theorem 2.1.8 pp. 74–75, or in [7], Theorem 13.Since h might not be a simple h Lie -module, we cannot just apply Schur’s lemma to the group Hom U ( h Lie ) ( h , N ) . But this is where the short exact sequence (2) comes in handy. As a sequenceof left h Lie -modules it actually splits, yielding the decomposition h = Leib ( h ) ⊕ h Lie and since h is a simple Leibniz algebra, this is the decomposition of h into simple h Lie -modules.Now, since N is also a simple h Lie -module, we get that if N ≃ Leib ( h ) or N ≃ h Lie (as a left h Lie -module), then H ∗− ( h Lie , k ) ⊗ Hom U ( h Lie ) ( h , N ) ≃ H ∗− ( h Lie , k ) If this is not the case, then H ∗− ( h Lie , k ) ⊗ Hom U ( h Lie ) ( h , N ) ≃ • Case 4: M is a nontrivial simple antisymmetric h -bimodule, and N is a simple symmetric h -bimodule.Using Theorem -1.2, we have HL q ( h , N s ) = 0 for q ≥ . Moreover, because HL ( h , N s ) = N h isa trivial h -bimodule, and since we can identify Hom ( M, HL ( h , N s )) ≃ M ⋆ ⊗ N h with the directsum of dim( N h ) copies of M ⋆ we find that H p ( h Lie , Hom ( M, HL ( h , N a ))) ≃ H p ( h Lie , M ⋆ ) ⊕ ... ⊕ H p ( h Lie , M ⋆ ) = 0 , since M being a simple nontrivial h Lie -module, so is M ⋆ . Thus yielding: Ext ∗ UL ( h ) ( M a , N s ) = 0 • Case 5: M is a nontrivial simple antisymmetric representation, and N is simple and antisymmetric.Here, Theorem -1.2 apply again, and we have that HL q ( h , N a ) = 0 only when q ∈ { , } . We checkthat HL ( h , N a ) is a trivial left h -module. By definition of the chain complex defining Leibnizcohomology, we have that CL ( h , N a ) = Hom ( h , N ) . Now for a morphism ϕ ∈ Hom ( h , N ) to beannihilated by the differential dL means satisfying: dL ϕ ( x, y ) := [ x, ϕ ( y )] L − ϕ ([ x, y ]) = 0 ∀ x, y ∈ h , which is exactly to say that the left action of h on the module Hom ( h , N ) is trivial. Therefore,the same arguments used in Case 4 still apply, and we get that E pq = 0 for q > , and: Ext ∗ UL ( h ) ( M a , N a ) = 0 • Case 6: M is a nontrivial simple symmetric representation, and N = k is the trivial h -bimodule.We apply Proposition 2.2 to k to find: Ext iUL ( h ) (( U ( h Lie ) s , k ) ≃ if i > 1 ≃ h if i = 1 ≃ k if i = 0because since in this case, the map f in Proposition 2.2 is zero. We can now plug this in the secondspectral sequence of Proposition 2.1, with X = k , and Z = M , to obtain: Ext ∗ UL ( h ) ( M s , k ) ≃ H ∗− ( h Lie , Hom ( M, h )) ≃ H ∗− ( h Lie , k ) ⊗ Hom U ( h Lie ) ( M, h ) Using the same arguments as in Case 3, we get that if M ≃ Leib ( h ) or M ≃ h Lie , then H ∗− ( h Lie , k ) ⊗ Hom U ( h Lie ) ( M, h ) ≃ H ∗− ( h Lie , k ) If this is not the case, then H ∗− ( h Lie , k ) ⊗ Hom U ( h Lie ) ( M, h ) ≃ • Case 7: M and N are both simple nontrivial symmetric h -bimodules.Applying Proposition 2.2 to N s , and because N is a symmetric h -bimodule, we find that Ext iUL ( h ) ( U ( h Lie ) s , N s ) ≃ i ≥ ≃ N if i = 0Now using the second spectral sequence of Proposition 2.1, we get: Ext ∗ UL ( h ) ( M s , N s ) ≃ H ∗ ( h Lie , Hom ( M, N )) ≃ H ∗ ( h Lie , k ) ⊗ Hom h Lie ( M, N ) Once again, since M and N are simple h Lie -modules, this vector space is nonzero only if M ≃ N ,in which case it is isomorphic to H ∗ ( h Lie , k ) . • Case 8: M is a simple nontrivial symmetric h -bimodule, and N is a simple nontrivial antisymmetric h -bimodule.By Proposition 2.2, we have: Ext iUL ( h ) ( U ( h Lie ) s , N a ) ≃ for i > 2 ≃ Hom ( h , Hom U ( h Lie ) ( h , N )) for i = 2 ≃ Coker ( h ) for i = 1 ≃ Ker ( h ) for i = 0The map h appearing here (defined in the statement of the theorem) is due to the fact that N isan antisymmetric h -bimodule. Moreover, since N is supposed to be nontrivial and h is a h -modulehomomorphism, Ker ( h ) = 0 . Therefore we have that E pq = 0 for q > and q = 0 . For theremaining values of q , we have isomorphisms E p ≃ H p ( h Lie , Hom ( M, b N )) ≃ H p ( h Lie , k ) ⊗ Hom U ( h Lie ) ( M, b N ) and E p ≃ H p ( h Lie , Hom ( M, Hom ( h , Hom U ( h Lie ) ( h , N )))) ≃ H p ( h Lie , k ) ⊗ Hom U ( h Lie ) ( M, Hom ( h , Hom U ( h Lie ) ( h , N ))) XT GROUPS IN THE CATEGORY OF BIMODULES OVER A SIMPLE LEIBNIZ ALGEBRA 11 The first isomorphism tells us that Ext UL ( h ) ( M s , N a ) ≃ Hom U ( h Lie ) ( M, b N ) .To use the second isomorphism, we need to proceed as in Case 6, since h is not a priori a simple h Lie -module, although with some more cases. – If N Leib ( h ) or N h Lie :Then Hom U ( h Lie ) ( h , N ) ≃ , yielding E p = 0 – If N ≃ Leib ( h ) or N ≃ h Lie :Then Hom U ( h Lie ) ( h , N ) ≃ k , and we have E p ≃ H p ( h Lie , Hom ( M, h ⋆ )) ≃ H p ( h Lie , k ) ⊗ Hom U ( h Lie ) ( M, h ⋆ ) Now we need to do the same work for Hom U ( h Lie ) ( M, h ⋆ ) . Since h is a simple Leibniz algebra, h Lie is a simple Lie algebra and is isomorphic as an h Lie -module to its dual h ⋆Lie via the Killingform. Moreover, the exactness of the functor Hom ( − , k ) gives us the short exact sequence −→ h ⋆Lie −→ h ⋆ −→ Leib ( h ) ⋆ −→ and the decomposition of h ⋆ = Leib ( h ) ⋆ ⊕ h ⋆Lie as a left h Lie -module, since the Leib ( h ) is asimple h Lie -module and therefore so is its dual.We therefore are in one of the following cases: ∗ If M Leib ( h ) ⋆ or M h ⋆Lie :Then Hom U ( h Lie ) ( M, h ⋆ ) ≃ , and E p = 0 . ∗ If M ≃ Leib ( h ) ⋆ or M ≃ h ⋆Lie :Then Hom U ( h Lie ) ( M, h ⋆ ) ≃ k , and we get E p ≃ H p ( h Lie , k ) ⊗ Hom U ( h Lie ) ( M, h ⋆ ) ≃ H p ( h Lie , k ) In order to get the promised vanishing of the Ext groups, we just use the fact that, since h Lie is simple: H ( h Lie , k ) ≃ H ( h Lie , k ) ≃ and this concludes our proof. (cid:3) This theorem also allows us to compute the Ext dimension of the category, denoted by L ( h ) in [10],of finite dimensional bimodules. First we need to make sure that the Ext groups are well defined in thiscategory. To do so we will use more general results from Category Theory.Notice that L ( h ) is an essentially small abelian category. This means by definition that the class ofisomorphism classes of objects is a set. This property implies that every one of its categories of fractionsexists and is essentially small as well (see for example Proposition 5.2.2 of [1]). Therefore the derivedcategory of L ( h ) is well defined.All there is to do now is to see that one can relate the Ext groups with morphisms in the derivedcategory. For this, we refer the reader to §5 of Chapter III, and §6.14 of Chapter III in [6]. In §5 theauthors define the Ext functor in terms of the Hom in the derived category, and they show in §6.14 thatit is equivalent to the derived functor definition.Therefore, the Ext groups are well defined in the category L ( h ) and we can now state the followingcorollary to Theorem 2.3: Corollary 2.4. For i ∈ { , , } , the natural transformation Ext iL ( h ) −→ Ext iUL ( h ) induced by the inclu-sion functor from L ( h ) to the category of U L ( h ) -modules is an isomorphism. Moreover, since Ext iL ( h ) = 0 for i > , the Ext dimension of L ( h ) is . Proof. For finite dimensional bimodules M and N , it is clear that Hom L ( h ) ( M, N ) = Hom UL ( h ) ( M, N ) ,as well as Ext L ( h ) ( M, N ) = Ext UL ( h ) ( M, N ) . This gives us the i ∈ { , } cases.For i = 2 , because of the short exact sequence (3), it is enough to consider the case when M and N are simple objects. In this case Theorem 2.3 tells us that Ext UL ( h ) ( M s , N a ) = 0 only when M ∈ { Leib ( h ) ⋆ , h ⋆Lie } and N ∈ { Leib ( h ) , h Lie } . When this Ext group is zero, then Ext L ( h ) −→ Ext UL ( h ) isobviously an isomorphism.When Ext UL ( h ) ( M s , N a ) = 0 , then it is one dimensional. This means that we only have to producea non trivial two-fold extension −→ N a −→ E −→ F −→ M s −→ with dim( E ) , dim( F ) < ∞ toconclude.Since Ext UL ( h ) ( M s , N a ) is one dimensional, then we can select one of its generators. It is the equivalenceclass of an exact sequence N a E F M s ϕ We can split this sequence into two exact sequences of length , −→ N a −→ E −→ Im ( ϕ ) −→ and −→ Im ( ϕ ) −→ F −→ M s −→ .Since the sequence represents a generator of Ext UL ( h ) ( M s , N a ) , the -fold exact sequences cannot split,i.e. represent trivial classes in the corresponding Ext UL ( h ) . But by Theorem 2.3 again, for this to be thecase, they must be in Ext UL ( h ) ( M s , k ) and Ext UL ( h ) ( k, N a ) respectively, since the bimodule Im ( ϕ ) mustbe both symmetric and antisymmetric (else the extensions are split, again using Theorem 2.3).But we know that Ext UL ( h ) ( M s , k ) = Ext L ( h ) ( M s , k ) , and Ext UL ( h ) ( k, N a ) = Ext L ( h ) ( k, N a ) . Thismeans that E and F must be of finite dimension, allowing us to conclude.The last step is to show that Ext iL ( h ) ( M, N ) = 0 for i ≥ . Note that this is not true for the wholecategory of U L ( h ) -bimodules. But we have seen in Theorem 2.3 that all higher Ext groups come fromhigher Chevalley-Eilenberg cohomology H ∗ ( h Lie , k ) = Ext ∗ U h Lie ( k, k ) of the simple Lie algebra h Lie . Weclaim that these Ext groups vanish for ∗ > in the subcategory (of finite dimensional h Lie -modules ofthe subcategory) of finite dimensional U L ( h ) -bimodules. Indeed, for a simple Lie algebra h , the categoryof finite-dimensional h Lie -modules is semisimple by Weyl’s theorem (see for example Theorem 7.8.11 in[13]). (cid:3) References [1] Borceux, Francis, Handbook of categorical algebra. 1. Basic category theory. Encyclopedia of Mathematics and its Ap-plications, Cambridge University Press, Cambridge, 1994[2] H. Cartan, S. Eilenberg, Homological algebra. With an appendix by David A. Buchsbaum. Reprint of the 1956 original.Princeton Landmarks in Mathematics. Princeton University Press, Princeton, NJ, 1999.[3] J. Feldvoss, Leibniz algebras as non-associative algebras. in: Nonassociative Mathematics and Its Applications, Denver,CO, 2017 (eds. P. Vojtěchovský, M. R. Bremner, J. S. Carter, A. B. Evans, J. Huerta, M. K. Kinyon, G. E. Moorhouse,J. D. H. Smith), Contemp. Math., vol. , Amer. Math. Soc., Providence, RI, 2019, pp. 115–149.[4] J. Feldvoss, F. Wagemann, On Leibniz cohomology arXiv:1902.06128 [5] D. B. 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Weibel, An introduction to homological algebra Cambridge studies in advanced mathematics , Cambridge UniversityPress 1995 Laboratoire de Mathématiques, Université Louvain-la-Neuve, Chemin du Cyclotron, 1348 Louvain-la-Neuve, Belgique E-mail address : [email protected] Laboratoire de mathématiques Jean Leray, UMR 6629 du CNRS, Université de Nantes, 2, rue de laHoussinière, F-44322 Nantes Cedex 3, France E-mail address ::