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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]

**Received date:** September 19, 2006

**Visit for more related articles at** Journal of Generalized Lie Theory and Applications

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.

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 c_{1} and c_{2} 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 Δ_{L} : defined by is a Lie coalgebra comultiplication, that is, if Δ_{L} 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.

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

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.

*G _{i}*-associative algebras

The class of associative Lie-admissible algebras contains interesting subclasses associated to the subgroups *G _{i}*of that we naturally call

**Definition 3. **Let *G _{i}* be a subgroup of The algebra is

**Proposition 3.** *Every G _{i} -associative algebra is a -associative algebra.*

**Proof.** Every subgroup *G _{i}* of corresponds to an invariant linear space generated by a single vector More precisely we consider

**Proposition 4.** Every *G _{i}*-associative algebra is a Lie-admissible algebra.

Proof. The vector V belongs to the orbits for every vi. Thus, if μ is a *G _{i}*-associative product, it also satisfies

and μ is a Lie-admissible multiplication.

We deduce the following type of Lie-admissible algebras:

1. A *G _{1}*-associative algebra is an associative algebra.

2. A *G _{2}*-associative algebra is a Vinberg algebra. If A is finite-dimensional, the associated Lie algebra is provided with an affine structure.

3. A *G _{3}*-associative algebra is a pre-Lie algebra.

4. If is *G _{4}*-associative then μ satisfies

with X · Y = μ(X, Y ).

5. If is *G _{5}*-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.

*G _{i}*-coalgebras

Dualizing the formula (2.3) we obtain the notion of *G _{i}*-coalgebra.

Definition 4. A *G _{i}*-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 *G _{i}*-associative algebra. A

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

2. (Un) The following diagram is commutative:

The axiom (*G _{i}*-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 *G _{i}*-coalgebra is a vector space M provided with a comultiplication Δ: and a counit such that

1. (*G _{i}*-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 *G _{i}*-coalgebras

is a linear map from M to M' such that

**Proposition 5.** Every *G _{i}*-coalgebra is a Lie-admissible coalgebra.

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

**The dual space of a G_{i}-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 *G _{i}*-coalgebra is provided with a structure of

**Proof.** Let (M,Δ) be a *G _{i}*-coalgebra. We consider the multiplication on the dual vector space M of M defined by It provides M with a

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 *G _{i}*-associative algebra has a

**Proof. **Let A be a finite dimensional *G _{i}*-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 *G _{i}*-associative coalgebra.

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 *G _{i}*-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 *G _{i}* − Ass be the quadratic operad associated to the

**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 *G _{i}*-associative algebra.

**Proposition 8. **Let (A, μ) be a *G _{i}*-associative algebra and (M, Δ) a G! i-coalgebra. Then the algebra (Hom(M,A), *) is a G

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.

**Lie-admissible bialgebras**

**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 *G _{i}*-bialgebras. As example, a compatibility condition for pre-Lie bialgebras (that is G3-bialgebras) is given by

**Tensor product of G_{i} 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 *G _{i}*-associative algebra and B a G! i-algebra (with the same index) then can be provided with a

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 G_{i}-coalgebras**

Let (M_{1},Δ_{1}) and (M_{2},Δ_{2}) be two Lie-admissible coalgebras and Δ the composite

If Δ_{1} is a comultiplication of *G _{i}*-coalgebra, what should be the structure of (M2,Δ

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!.

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

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- GozeM, RemmE (2004) Lie-admissible algebras and operads. J Algebra 273: 129-152.
- GozeM, RemmE (2007) A class of nonassociative algebras. Algebra Colloq.
- GozeM, RemmE (2006) The quadratic operad ̃P and tensor products of algebras. Preprint, arXiv: math.RA/0606105.
- MyungHC (1982) Lie algebras and Flexible Lie-admissible Algebras.Hadronic Press.
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