﻿ Canonical Form Associated with an r-Jacobi Algebra

Advances in Linear Algebra & Matrix Theory
Vol.06 No.01(2016), Article ID:64395,5 pages
10.4236/alamt.2016.61003

Canonical Form Associated with an r-Jacobi Algebra

Deva Michelle Bouaboté Ntoumba

Faculty of Science and Technology, Marien NGOUABI University, Brazzaville, Congo    Received 13 November 2015; accepted 7 March 2016; published 10 March 2016

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.

Keywords:

Module of Kähler Differential, Lie Algebra of Order r, Jacobi Algebra of Order r 1. Introduction

The concept of n-Lie algebra over a field K, n an integer ≥2, introduced by Fillipov  , 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   .

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   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   .

2. Structure of Jacobi Algebra of Order r ≥ 1

A-Module A × ΩK(A)

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

The pair has the following universal property    : 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  .

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. 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.

Cite this paper

Deva Michelle BouabotéNtoumba, (2016) Canonical Form Associated with an r-Jacobi Algebra. Advances in Linear Algebra & Matrix Theory,06,17-21. doi: 10.4236/alamt.2016.61003

References

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

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