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Cup-product for equivariant Leibniz cohomology and zinbiel algebras

We study finite group actions on Leibniz algebras, define equivariant cohomology groups associated to such actions. We show that there exists a cup-product operation on this graded cohomology groups which makes it a graded zinbiel algebra.


Introduction
In [10], J.-L. Loday introduced some new types of algebras along with their (co)homologies and studied the associated operads. Leibniz algebras and their Koszul duals, zinbiel algebras are examples of such algebras. Let Leib be the category of Leibniz algebras over a fixed field K. Given any Leibniz algebra g, J.-L. Loday [9] introduced a cup-product operations ∪ : HL p (g; A) × HL q (g; A) → HL p+q (g; A) on the graded Leibniz cohomology HL * (g; A) groups with coefficients in a commutative, associative algebra A. This product is neither associative nor commutative, but satisfies the formula which is, the defining relation of a zinbiel algebra. Thus, the author proved that HL * (g; A) is a graded zinbiel algebra. The aim of this paper is to study finite group actions on Leibniz algebras. Let G be a finite group and g be a Leibniz algebra equipped with a given action of G. We discuss examples of such actions and introduce equivariant cohomology groups of a Leibniz algebra g equipped with an action of a finite group G, along the line of Bredon cohomology of a G-space [2]. We introduce a cup-product operation in the equivariant context and prove that for a Leibniz algebra g equipped with an action of G, equivariant graded Leibniz cohomology groups also admit a graded zinbiel algebra structure.

Preliminaries
In this section, we recall some definitions, notations and results from [8], [9], [10]. In [6], J.-L. Loday observed that for a Lie algebra g if one replaces the exterior product ∧ by the tensor product ⊗ in the classical formula for the boundary map d of the Chevalley-Eilenberg complex and modifies the boundary map d so as to put the commutator [x i , x j ] at the place i when i < j (see 10.6.2.1, [6]), then one obtains a new complex (T g, d). The only relation that is used to , y] for x, y, z ∈ g.
Dualizing this complex one gets the Leibniz cohomology of the Lie algebra g. Thus, Leibniz cohomology is defined for a larger class of algebras: the Leibniz algebras. More explicitly, it is defined as follows.
2.1. Definition. Let K be a field. A Leibniz algebra is a vector space g over K, equipped with a bracket operation, which is K-bilinear and satisfies the Leibniz identity: A graded Leibniz algebra is a graded K-vector space g = {g i } i≥0 , equipped with a graded bracket operation of degree 0 which is K-bilinear and satisfies the graded Leibniz identity: , y] for homogeneous elements x, y, z ∈ g.
Any Lie algebra is automatically a Leibniz algebra, as in the presence of skew symmetry, the Jacobi identity is equivalent to the Leibniz identity.
2.2. Example. Let (g, d) be a differential Lie algebra with the Lie bracket [, ]. Then g is a Leibniz algebra with the bracket operation [x, y] d := [x, dy]. The new bracket on g is called the derived bracket.

2.3.
Example. Consider a three dimensional vector space g spanned by {e 1 , e 2 , e 3 } over C. Define a bilinear map [ , ] : g × g −→ g by [e 1 , e 3 ] = e 2 and [e 3 , e 3 ] = e 1 , all other products of basis elements being 0. Then (g, [ , ]) is a Leibniz algebra over C of dimension 3. The Leibniz algebra g is nilpotent and is denoted by λ 6 in the classification of three dimensional nilpotent Leibniz algebras, see [1,7].
of Leibniz algebras is a Linear map which preserves the brackets, that is, Recall that the homology HL * (g) of a Leibniz algebra is defined as follows. To any Leibniz algebra g there is an associated chain complex The map d satisfy d 2 = 0 ( [6]). The homology groups of this complex are denoted by HL n (g), n ≥ 1.
Next, recall that the Leibniz cohomology HL * (g; A) of a Leibniz algebra g with coefficients in an associative commutative K-algebra A is defined as follows.

Group actions on Leibniz algebras
The purpose of this section is to introduce finite group actions on Leibniz algebras and provide examples of group actions.
3.1. Definition. Let g be a Leibniz algebra and G be a finite group. The group G is said to act from the left if there exists a function φ : G × g → g, (g, x) → φ(g, x) = gx satisfying the following conditions.
(1) For each g ∈ G the map x → gx, denoted by ψ g is linear.
(2) ex = x for all x ∈ g, where e ∈ G is the group identity.
When g = {g i } i≥0 is a graded Leibniz algebra, we further assume that for each g ∈ G the map ψ g is graded linear of degree 0.
The following is an equivalent formulation of the above definition.

3.2.
Proposition. Let G be a finite group and g be a Leibniz algebra. Then G acts on g if and only if there exists a group homomorphism ψ : G → Iso Leib (g, g), g → ψ(g) = ψ g from the group G to the group of Leibniz algebra isomorphisms from g to g, where ψ g (x) = gx is the left translation by g.

3.3.
Remark. Let K[G] be the group ring. If a Leibniz algebra g is equipped with an action of G then g may be viewed as a K[G]-module.
Observe that if g is a Leibniz algebra equipped with an action of a group G as above, then for every subgroup H ⊂ G, the H-fixed point set g H is defined by Clearly, for every subgroup H ⊂ G, g H is sub Leibniz algebra of g. Moreover, note that if H, K are subgroups of G with g −1 Hg ⊂ K, g ∈ G, then the Leibniz algebra homomorphism ψ g maps g K to g H .

3.4.
Example. Let V be K-module which is a representation space of a finite group G. On there is a unique bracket that makes it into a Leibniz algebra and verifies This is the free Leibniz algebra over the K-module V . The linear action of G on V extends naturally to an action onT (V ) and the bracket defined above satisfies the conditions of Definition 3.1. Thus, (G,T (V )) is an action of G on the free Leibniz algebraT (V ).

3.5.
Example. Let (g, d) be a differential Lie algebra with the Lie bracket [ , ]. Assume that a finite group G acts linearly on g such that (1) [gx, gy] = g[x, y] for all g ∈ G and x, y ∈ g, (2) d(gx) = gd(x) for all g ∈ G and x ∈ g.
Then, the group G acts on the Leibniz algebra (g, Our next example is based on a specific case of the above Example.
3.6. Example. Let V be a vector space over a field K. Assume that a finite group G acts linearly and let [α, β] ∈ C k+l−1 (V ) be the Gerstenhaber bracket defined by: Next observe that the linear action of G on V induces an action on C k (V ) for each k and is given by Thus, an element α ∈ C k (V ) is invariant with respect to the above action if and only if α : V × · · · × V → V is an equivariant multilinear map where G acts component wise on any product of V. Moreover, it is straight forward to verify that the above action on the graded vector space Assume that there is an element α ∈ C 2 (V ) which is invariant with respect to the above action and defines an associative algebra structure on V . The last assertion is equivalent to assume that [α, α] = 0. For such an element α ∈ C 2 (V ) we define a map d α : Then, (C • (V ), [ , ] dα ) is a graded Leibniz algebra equipped with an action of the group G.
The following discussion is a prelude to our next example.
3.8. Definition. Let G be a finite group and M is a smooth G-manifold. Then for any k ≥ 0, there is an action of G on the space of k-forms Ω k (M ) on M , given by for α ∈ Ω k (M ) and x ∈ M . Moreover, there is an action of G on the space of vector fields X (M ) on M , namely, for X ∈ X (M ) and x ∈ M . This action is the same as the action of G on χ(M ) by push forward by l g as discussed in the above example.
3.9. Lemma. The contraction operator and the de Rham differential operator satisfies the following properties. For any α ∈ Ω k (M ), X ∈ X (M ) and g ∈ G, Proof. (i) For any x ∈ M and X 1 , . . . , X k−1 ∈ T x M , we have (ii) Note that the action of G on C ∞ (M ) is given by (g, f ) := (l g −1 ) * f . Therefore, for any x ∈ M , The result now follows from the observation that for any α, β ∈ Ω • (M ), (iii) It follows from part (i) and (ii) and the Cartan magic formula (iv) For any x ∈ M and β x ∈ T * x M , we have Recall that a Nambu-Poisson manifold is a generalization of the notion of Poisson manifolds and is defined as follows [11], [4]. 3.11. Remark. Recall [11] that Poisson manifolds are Nambu-Poisson manifolds of order 2.

Definition. A (left) Leibniz algebroid over a smooth manifold M is a smooth vector bundle
We recall the following result from [12], [4].
Let M be a Nambu-Poisson manifold of order n with the corresponding Nambu tensor Π. Then the bundle n−1 T * M carries a Leibniz algebroid structure with bracket on the space Ω n−1 (M ) of (n − 1)-forms on M and the anchor is given by the bundle map Π ♯ . Thus Ω n−1 (M ) is a Leibniz algebra with respect this bracket.
3.13. Example. Suppose G is a finite group. Let M be a Nambu-Poisson manifold of order n with the associated Nambu tensor Π. Assume that G acts smoothly on M and Π is G-invariant with respect to the action of G as defined in Definition 3.8. Then, it follows from Lemma 3.9 that for any α, β ∈ Ω n−1 (M ) and g ∈ G, Thus, with the bracket [ , ] as defined above Ω n−1 (M ) is a Leibniz algebra equipped with the action of the given group G.

Equivariant cohomology of a Leibniz algebra equipped with a group action
In this section we introduce equivariant cohomology groups of a Leibniz algebra equipped with an action of a finite group following [2], [5].
Let G be a finite group. Recall that the category of canonical orbits of G, denoted by O G , is a category whose objects are left cosets G/H, as H runs over the all subgroups of G. Note that the group G acts on the set G/H by left translation. A morphism from G/H to G/K is a G-map. Recall that such a morphism determines and is determined by a subconjugacy relation g −1 Hg ⊆ K and is given byĝ(eH) = gK. We denote this morphism byĝ [2] If g is a Leibniz algebra equipped with an action of G, then we have contravariant functor Φg : O G → Leib, given by Φg(G/H) = g H and for a morphismĝ : G/H → G/K corresponding to a sub conjugacy relation g −1 Hg ⊆ K, Φg(ĝ) = ψ g : g K → g H . Thus, Φg is an O G -Leibniz algebra, which will be referred to as the O G -Leibniz algebra associated to g.
We now proceed to define the notion of equivariant cohomology of a Leibniz algebra g equipped with an action of a finite group G.  We have an induced homomorphism δ : CL n G (g; A) → CL n+1 G (g; A) given by δ(c) := c • d for any natural transformation c ∈ CL n G (g; A). Thus, we have a cochain complex CL ♯ G (g; A) = {CL n G (g; A), δ}. The homology groups of this cochain complex are equivariant Leibniz algebra cohomology groups of (G, g) with coefficients in A and denoted by HL n G (g; A).
The following is an equivalent formulation of the groups HL n G (g; A). Set S n (g; A) = ⊕ H<G CL n (g H ; A(G/H)) and define δ : S n (g; A) → S n+1 (g; A) by δ = ⊕ H<G δ H , where δ H : CL n (g H ; A(G/H)) → CL n+1 (g H ; A(G/H)) is the non-equivariant coboundary map (2.2) for the Leibniz algebra g H . Clearly, {S n (g; A), δ} is a cochain complex. We define a subcomplex of this cochain complex as follows.

Definition.
A cochain c = {c H } ∈ S n (g; A) is said to be invariant under the action of G if for every morphismĝ : G/H → G/K, corresponding to a subconjugacy relation g −1 Hg ⊂ K following holds:

4.4.
Lemma. The set of all invariant n-cochains is a a subgroup S n G (g; A) of S n (g; A).
Proof. It is clear that S n G (g; A) is a subgroup of S n (g; A) as A(ĝ) is a homomorphism for everŷ g : G/H → G/K. Let c = {c H } ∈ S n (g; A) be invariant andĝ : G/H → G/K be a morphism in O G corresponding to a subconjugacy relation g −1 Hg ⊂ K . Thus, for every (x 1 , . . . , x n ) ∈ (g K ) ⊗n we have c H (ψ g (x 1 ), . . . , ψ g (x n )) = A(ĝ)(c K (x 1 , . . . , x n )).

4.5.
Theorem. Let g be a Leibniz algebra with a given action of G. For any O G -algebra A we have an isomorphism H n (S ♯ G (g; A)) ∼ = HL n G (g; A) for all n.
Proof. Let F ∈ CL n G (g; A). Note that the collection {F (G/H) : (g H ) ⊗n → A(G/H)} of nonequivariant n-cochain of g H , as H varies over subgroups of G, is invariant. Because, for every morphismĝ : G/H → G/K of O G , we have A(ĝ) • F (G/K) = F (G/H) • (ψ g ) ⊗n by naturality of F. Thus, we have a map α n : CL n G (g; A) → S n G (g; A) defined by F → α n (F ) := {F (G/H)} for all n. We claim that α = {α n } is a cochain map. Let Thus, α induces a homomorphismᾱ n : HL n G (g; A) → H n (S ♯ G (g; A)) for all n.
Finally, observe that the cochain map α = {α n } is a cochain isomorphism with inverse β = {β n } defined as follows: β : S n G (g; A) → CL n G (g; A), {c H } → C, where the natural transformation C : CL n (g) → A is given by C(G/H) := c H . The naturality of C follows from the invariance of {c H }. Thus,ᾱ n is an isomorphism for all n.

Equivariant Leibniz cohomology as zinbiel algebra
In [9], J.-L. Loday introduced zinbiel algebras and proved that for any Leibniz algebra g the graded Leibniz cohomology with coefficients in a commutative, associative algebra admits a graded product which makes it a graded zinbiel algebra. The aim of this section is to prove an equivariant version of this result. Explicitly, for a Leibniz algebra g equipped with an action of G, we prove that equivariant graded Leibniz cohomology HL * G (g, A) as introduced in the previous section also admits a graded zinbiel algebra structure. To show this, we use the equivalent formulation (Theorem 4.5) of equivariant cohomology to define a cup-product operation which is based on the product defined at the cochain level of the fixed points Leibniz algebras {g H } H<G .
In the group algebra K[S n ], let sh p,q be the element where the summation is over all (p, q)-shuffles.
For any vector space V we let σ ∈ S n act on V ⊗n by where the generator v 1 ⊗ · · · ⊗ v n of V ⊗n is denoted by v 1 . . . v n . The free Zinbiel algebra over the vector space V isT (V ) = ⊕ n≥1 V ⊗n equipped with the following product Here Id 1 is the identity on the first factor. Note that the linear map from K[S n ] to itself induced by σ → sgn(σ)σ −1 for σ ∈ S n is an anti-homomorphism. Let us denote the image of α ∈ K[S n ] under this map byα.
Let g be a Leibniz algebra equipped with a given action of a finite group G. Let Φg be the corresponding O G -Leibniz algebra. Denote by Φg ⊗p+q the O G -vector space given by For any non-negative integers p and q we define a natural transformation ρ p,q := Id 1 ⊗ sh p−1,q = Φg ⊗p+q → Φg ⊗p+q . where the above sum is over all (p − 1, q)-shuffles σ.
Let τ p,q : Φg ⊗p+q → Φg ⊗p+q be the natural transformation defined as follows. For any object G/H in O G and for generators x = v 1 . . . v p ∈ (g H ) ⊗p and y = v p+1 . . . v p+q ∈ (g H ) ⊗q , τ p,q (G/H)(xy) = yx with the obvious definition on morphisms in O G . Then, for non-negative integers p, q, r, we have the following equality (ρ p,q ⊗ Id r ) • ρ p+q,r = (Id p ⊗ ρ q,r + (−1) rq • τ r,q • ρ r,q ) • ρ p,q+r In other words, Clearly, c ∪ d ∈ S n (g; A). We claim that c ∪ d ∈ S p+q G (g; A). Thus, we need to prove that c ∪ d is invariant. Letĝ : G/H → G/K, g −1 Hg ⊂ K be a morphism in O G . We need to check that Note that • ρ p,q (G/K) (by invariance of c and d) Next, note that the cup-product operation is well-defined. This is because,

5.3.
Theorem. Given a Leibniz algebra g equipped with an action of a finite group G and an O Galgebra A, the graded equivariant cohomology HL * G (g; A) is a graded zinbiel algebra with respect to the cup-product operation (5.2).