ISSN: 1736-4337
Journal of Generalized Lie Theory and Applications
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Michel GOZE* and Elisabeth REMM

Universit´e de Haute Alsace, F.S.T., 4 rue des Fr`eres Lumi`ere, 68093 Mulhouse, France

*Corresponding Author:
Michel GOZE
Universit´e de Haute Alsace, F.S.T., 4 rue des Fr`eres Lumi`ere, 68093 Mulhouse, France
E-mails: [email protected] and [email protected]

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#### Abstract

After introducing the concept of Lie-admissible coalgebras, we study a remarkable class corresponding to coalgebras whose coassociator satisfies invariance conditions with respect to the symmetric group 3. We then study the convolution and tensor products.

#### Definitions and examples

In this work indicates a field of characteristic zero. Let M be a -vector space and Δ a linear comultiplication map Δ: The coassociator of Δ is denoted by

and the flip is the linear map defined by

Let be the symmetric group of degree 3. We denote by c1 and c2 the 3-cycles of and the transposition echanging i and j. For every we define a linear map by

Definition 1. The pair (M, Δ) is a Lie-admissible coalgebra if the linear map ΔL : defined by is a Lie coalgebra comultiplication, that is, if ΔL satisfies

Recall that a multiplication μ of a algebra (A, μ) is Lie-admissible if its associator

satisfies

where ε(σ) is the sign of the permutation σ. This means that the algebra (A, [, ]) whose product is given by the bracket [x, y] = μ(x, y)−μ(y, x) is a Lie algebra. We have a similar characterization of a Lie-admissible comultiplication.

Proposition 1. A comultiplication Δ on M is a Lie-admissible comultiplication if and only if Δ satisfies

where ε(σ) denotes the sign of the permutation σ.

Proof. It is a direct consequence of Equation (1.1) because

This proves the proposition.

Examples

• Every coassociative coalgebra is a Lie-admissible coalgebra.

• The comultiplication of a pre-Lie coalgebra (M, Δ) satisfies

(1.3)

Since the composition of (1.3) by and gives respectively

and

we obtain Identity (1.2) by summation of (1.3) with these two equations and every pre-Lie coalgebra is Lie-admissible.

In the following sections we generalize these examples.

#### Gi-coalgebras

An interesting class of Lie-admissible coalgebras is obtained by dualizing the Gi-associative algebras. These Lie-admissible algebras has been introduced in [9] and developed in [3]. Let us point out these initially notations.

#### -associative algebras

Let be the group algebra associated to where is a field of characteristic zero. Every decomposes as follows:

or simply

where If A is a vector space, then we define from such a vector u the endomorphism by

Consider the natural right action of on

The corresponding orbit of a vector is denoted by and generates a linear subspace It is an invariant subspace of Therefore, using Mashke’s theorem, it is a direct product of irreducible invariant subspaces.

Let be a algebra with multiplication μ and A(μ) its associator.

Definition 2. An algebra is a associative algebra if there exists such that

Proposition 2. Let v be in such that dim Then with and the vectors V and W are the following vectors:

(2.1)

(2.2)

The first case corresponds to the character of 3 given by the sign, the second corresponds to the trivial case.

Every algebra whose associator satisfies

is a Lie-admissible algebra. Likewise an algebra whose associator satisfies

is 3-power associative, that is, it satisfies A(μ)(x, x, x) = 0 for every x 2 A.

Gi-associative algebras

The class of associative Lie-admissible algebras contains interesting subclasses associated to the subgroups Giof that we naturally call Gi associative algebras. Let us introduce some notations. Consider the subgroups of

Definition 3. Let Gi be a subgroup of The algebra is Gi-associative if

Proposition 3. Every Gi -associative algebra is a -associative algebra.

Proof. Every subgroup Gi of corresponds to an invariant linear space generated by a single vector More precisely we consider

Proposition 4. Every Gi-associative algebra is a Lie-admissible algebra.

Proof. The vector V belongs to the orbits for every vi. Thus, if μ is a Gi-associative product, it also satisfies

and μ is a Lie-admissible multiplication.

We deduce the following type of Lie-admissible algebras:

1. A G1-associative algebra is an associative algebra.

2. A G2-associative algebra is a Vinberg algebra. If A is finite-dimensional, the associated Lie algebra is provided with an affine structure.

3. A G3-associative algebra is a pre-Lie algebra.

4. If is G4-associative then μ satisfies

with X · Y = μ(X, Y ).

5. If is G5-associative then μ satisfies the generalized Jacobi condition :

with X · Y = μ(X, Y ). Moreover if the product is skew-symmetric, then it is a Lie algebra bracket.

6. A G6-associative algebra is a Lie-admissible algebra.

Gi-coalgebras

Dualizing the formula (2.3) we obtain the notion of Gi-coalgebra.

Definition 4. A Gi-coalgebra is a -vector space M provided with a comultiplication Δ satisfying

Remark. We can present an equivalent and axiomatic definition of the notion of Gi-associative algebra. A Gi-associative algebra is (A, μ, ,Gi) where A is a vector space, Gi a subgroup of are linear maps satisfying the following axioms:

If we impose that the algebra is unitary we have to add the following axiom:

2. (Un) The following diagram is commutative:

The axiom (Gi-ass) expresses that the multiplication μ is

-associative whereas the axiom (Un) means that the element (1) of A is a left and right unit for μ. We want to dualize the previous diagrams to obtain the notions of corresponding coalgebras. Let Δbe a comultiplication

by

A Gi-coalgebra is a vector space M provided with a comultiplication Δ: and a counit such that

1. (Gi-ass co) The following square is commutative:

If we suppose moreover that the coalgebra is counitary we have to add the following axiom:

2. (Coun) The following diagram is commutative:

A morphism of Gi-coalgebras

is a linear map from M to M' such that

Proposition 5. Every Gi-coalgebra is a Lie-admissible coalgebra.

Proof. The Lie-admissible coalgebras are given by the relation

The dual space of a Gi-coalgebra

For any natural number n and any -vector spaces E and F, we denote by

the natural embedding

Proposition 6. The dual space of a Gi-coalgebra is provided with a structure of Gi-associative algebra.

Proof. Let (M,Δ) be a Gi-coalgebra. We consider the multiplication on the dual vector space M of M defined by It provides M with a Gi-associative algebra structure. In fact we have

for all f1, f2 2 M* where μK is the multiplication in K. Equation (2.4) becomes:

Proposition 7. The dual vector space of a finite dimensional Gi-associative algebra has a Gicoalgebra structure.

Proof. Let A be a finite dimensional Gi-associative algebra and letbe a basis of A. If {fi} is the dual basis then is a basis of The coproduct Δ on A* is defined by

In particular

where are the structure constants of μ related to the basis Then Δ is the comultiplication of a Gi-associative coalgebra.

#### The convolution product

Let us recall that if is associative algebra and (M,Δ) a coassociative coalgebra (i.e.a G1-coalgebra) then the convolution product

provides Hom(M,A) with an associative algebra structure. This result can be extended to the Gi-associative algebras and coalgebras. But we have to introduce the notion of G! i-coalgebras defined by the Koszul duality in the operads theory [2] and [3].

The G! i-algebras and coalgebras

Let Gi − Ass be the quadratic operad associated to the Gi-associative algebras. In [3] and [8], we show that these operads satisfy the Koszul duality as soon as i = 1, 2, 3, 6. Let Gi −Ass! be the dual operad. We will call a G! i-algebra any algebra on Gi−Ass!. These algebras are defined as follows:

Definition 5. For i  2, a G! i-algebra is an associative algebra satisfying

Definition 6. For i  2, a G! i-coalgebra is a coassociative coalgebra satisfying

We will provide Hom(M,A) with a structure of Gi-associative algebra.

Proposition 8. Let (A, μ) be a Gi-associative algebra and (M, Δ) a G! i-coalgebra. Then the algebra (Hom(M,A), *) is a Gi-associative algebra where * is the convolution product

Proof. Let us compute the associator A(*) of the convolution product. Since

But

This gives

Since Δ is coassociative,

the G! i-coalgebra structure implies

Then

This proves the proposition.

Definition 7. A Lie-admissible bialgebra is a triple where is a Lie-admissible algebra and a Lie-admissible coalgebra with a compatibility condition between  and μ:

Here we do not assume that the algebra and coalgebra are unitary and counitary. Among Lieadmissible bialgebras, we shall have the class of Gi-bialgebras. As example, a compatibility condition for pre-Lie bialgebras (that is G3-bialgebras) is given by

#### Tensor product of Lie-admissible algebras and coalgebras

Tensor product of Gi and G! i-algebras

We know that the tensor product of associative algebras can be provided with an associative algebra structure. In other words, the category of associative algebras is monoidal and closed for the tensor product. This is not true in general for other categories of-associative algebras.

Proposition 9. Let andbe two associative algebras respectively defined by the relations A(μA) and A(μB) Then is a associative algebra if and only if A and B are associative algebras (i.e G1-associative algebras).

Proof. See [4].

But we have:

Theorem 10. If A is a Gi-associative algebra and B a G! i-algebra (with the same index) then can be provided with a Gi-algebra structure for i = 1, . . . , 6.

Proof. Let us consider on the classical tensor product

To simplify, we denote by μ the product is an associative algebra, the associator A(μ) satisfies

Therefore

This proves the proposition.

Tensor product of Gi-coalgebras

Let (M11) and (M22) be two Lie-admissible coalgebras and Δ the composite

If Δ1 is a comultiplication of Gi-coalgebra, what should be the structure of (M2,Δ2) such that Δ is a comultiplication of Gi-coalgebra too?

Remark. In [5] we have generalized this study and defined for any quadratic operad P a quadratic operad ˜ P so that the tensor product of a P-algebra with a ˜ P-algebra is provided with a P-algebra structure. In the previous case we have always ˜ P = P!.

#### Acknowledgement

This work has been supported for the two authors by AUF project MASI 2005-2006.

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