Canonical Form Associated with an r-Jacobi Algebra

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Received 13 November 2015; accepted 7 March 2016; published 10 March 2016

1. Introduction

The concept of n-Lie algebra over a field K, n an integer ≥2, introduced by Fillipov [1] , is a generalization of the concept of Lie algebra over a field K, which corresponds to the case where n = 2. A structure of n-Lie algebra over a K-vector space W, is the given of an alternating multilinear mapping of degree n

verifying the identity

for all. This identity is called Jacobi identity of n-Lie algebra w [1] [2] .

A derivation of an n-Lie algebra is a K-linear map

such that

for all.

The set of all derivations of a n-Lie algebra W is a K-Lie algebra denoted by.

If is a n-Lie algebra, then for all, the map

is a derivation of.

When A is a commutative algebra, with unit 1_{A} over a commutative field K of characteristic zero, and when M is a A-module, a linear map

is a differential operator of order ≤1 [3] [4] if, for all a and b belonging to A,

When, we have the usual notion of derivation from A into M.

We denote by the A-module of differential operator of order ≤1 from A into M and by the A-module of differential operator of order ≤1 on A (M = A).

The aim of this work is to define the notion of r-Jacobi algebra and to construct the canonical form associated with this r-Jacobi algebra.

In the following, A denotes a unitary commutative algebra over a commutative field K of characteristic zero with unit 1_{A} and the module of Kähler differential of A and

the canonical derivation [3] [4] .

2. Structure of Jacobi Algebra of Order r ≥ 1

A-Module A × Ω_{K}(A)

Proposition 1 [3] The map is a differential operator of order ≤1. Moreover the image of generates the A-module.

The pair has the following universal property [3] [5] [6] : for all A-module M and for all differential operator of order ≤1

there exists an unique A-linear map

such that

Moreover, the map

is an isomorphism of A-modules.

For all integer, we say that an alternating K-multilinear map

is a alternating p-differential operator if for all, the map

is a alternating differential operator of order ≤ 1 for all.

We denote by, the A-module of alternating A-multilinear maps of degree p from into M and, the A-module of alternating p-differential operators from A into M.

One notes

such that

for all.

When is the A-exterior algebra of the A-module the differential operator

can be extended into a differential operator again noted

of degree +1 and of square 0. Thus, the pair is a differential complex [3] .

For all A-module M and for all alternating p-differential operator

there exists an unique alternating A-multilinear map of degree p

such that

Thus, the existence of an unique A-linear map

such that

for all elements of A when the map

is a alternating p-differential operator. Moreover, the map

is an isomorphism of A-modules [3] .

3. Structure of r-Jacobi Algebra

We say that a commutative algebra with unit A on a commutative field K of characteristic zero, is a r-Jacobi algebra, an integer, if A is provided with a structure of 2r-Lie algebra over K of bracket such that for all the map

is a differential operator of order ≤1.

Proposition 2 When A is a r-Jacobi algebra, then there exist an unique A-linear map

such that, for all

Proof. The map

is an alternating -differential operator. Thus deduced the existence and the uniqueness of the A-linear map

such that

That ends the proof.

Canonical form Associated with a r-Jacobi Algebra

In what follows, A is a r-Jacobi algebra.

Theorem 3 The map

is an alternating 2r-differential operator and induces an alternating A-multilinear mapping and only one of degree 2r

such that

Proof. As the map

is a A-differential operator of order ≤ 1 and the map

is an alternating -differential operator.

The unique A-alternating multinear map of degree 2r

induce an unique A-linear map

such that

for all

We say that is the canonical form associated with the r-Jacobi algebra A.

Corollary 1 For all

for any.

Acknowledgements

The author thanks Prof. E. Okassa for his remarks and sugestions.

References

[1] Fillipov, V.T. (1985) N-Lie Algebra. Sib. Mat. J., 26, 126-140.

[2] Bossoto, B.G.R., Okassa, E. Omporo, M. Lie algebra of an n-Lie algebra. Arxiv: 1506.06306v1.

[3] Okassa, E. (2007) Algèbres de Jacobi et algèbres de Lie-Rinehart-Jacobi. Journal of Pure and Applied Algebra, 208, 1071-1089.

http://dx.doi.org/10.1016/j.jpaa.2006.05.013

[4] Okassa, E. (2008) On Lie-Rinehart-Jacobi Algebras. Journal of Algebras and Its Aplications, 7, 749-772.

[5] Bourbaki, N. (1970) Algèbre. Chapitres 1 à 3. Hermann, Paris.

[6] Bourbaki, N. (1980) Algèbre. Chapitre 10, Algèbre Homologique. Masson, Paris, New York, Barcelone, Milan.