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Canonical Form Associated with an r-Jacobi Algebra

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DOI: 10.4236/alamt.2016.61003    2,535 Downloads   3,028 Views  

ABSTRACT

In this paper, we denote by A a commutative and unitary algebra over a commutative field K of characteristic 0 and r an integer ≥1. We define the notion of r-Jacobi algebra A and we construct the canonical form associated with the r-Jacobi algebra A.

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 1A 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 1A 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 × ΩK(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.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

Ntoumba, D. (2016) Canonical Form Associated with an r-Jacobi Algebra. Advances in Linear Algebra & Matrix Theory, 6, 17-21. doi: 10.4236/alamt.2016.61003.

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.

  
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