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ISSN: 1736-4337
Journal of Generalized Lie Theory and Applications
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Lie-admissible coalgebras

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]

Received date: September 19, 2006

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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 imageindicates a field of characteristic zero. Let M be a image-vector space and Δ a imagelinear comultiplication map Δ: image The coassociator of Δ is denoted by

image

and the flip image is the linear map defined byimage

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

image

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

image

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

image

satisfies

image

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

image

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

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

image

This proves the proposition.

Examples

• Every coassociative coalgebra is a Lie-admissible coalgebra.

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

image(1.3)

Since the composition of (1.3) by imageand image gives respectively

image

and

image

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.

image-associative algebras

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

image

or simply

image

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

image

Consider the natural right action of image on image

image

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

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

Definition 2. An algebra image is a image associative algebra if there existsimage such thatimage

Proposition 2. Let v be in image such that dimimage Thenimage withimage and the vectors V and W are the following vectors:

image(2.1)

image(2.2)

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

Every algebraimage whose associator satisfies

image

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

image

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

image

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

image

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

Proof. Every subgroup Gi of image corresponds to an invariant linear spaceimage generated by a single vectorimage More precisely we considerimageimageimage

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

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

image

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 image is G4-associative then μ satisfies

image

with X · Y = μ(X, Y ).

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

image

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 image-vector space M provided with a comultiplication Δ satisfying

image

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 image are linear maps satisfying the following axioms:

image

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

2. (Un) The following diagram is commutative:

image

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

image

by

image

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

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

image

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

2. (Coun) The following diagram is commutative:

image

A morphism of Gi-coalgebras

image

is a linear map from M to M' such that

image

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

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

image

image

The dual space of a Gi-coalgebra

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

image

the natural embedding

image

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 image It provides M with a Gi-associative algebra structure. In fact we have

image

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

image

image

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 letimagebe a basis of A. If {fi} is the dual basis then imageis a basis of image The coproduct Δ on A* is defined by

image

In particular

image

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

The convolution product

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

image

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

image

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

image

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

image

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

image

image

image

But

image

This gives

image

Since Δ is coassociative,

image

the G! i-coalgebra structure implies

image

Then

image

This proves the proposition.

Lie-admissible bialgebras

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

image

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

image

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 ofimage-associative algebras.

Proposition 9. Let image andimagebe two image associative algebras respectively defined by the relations A(μA) imageand A(μB) imageThen image is a image 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 imagecan be provided with a Gi-algebra structure for i = 1, . . . , 6.

Proof. Let us consider onimage the classical tensor product

image

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

image

Therefore

image

image

This proves the proposition.

Tensor product of Gi-coalgebras

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

image

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?

image

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.

References

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