_{1}

^{*}

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.

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

verifying the identity

for all

A derivation of an n-Lie algebra

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 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 [

When

We denote by

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 canonical derivation [

_{K}(A)

Proposition 1 [

The pair

there exists an unique A-linear map

such that

Moreover, the map

is an isomorphism of A-modules.

For all integer

is a alternating p-differential operator if for all

is a alternating differential operator of order ≤ 1 for all

We denote by

One notes

such that

for all

When

can be extended into a differential operator again noted

of degree +1 and of square 0. Thus, the pair

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

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

is an isomorphism of A-modules [

We say that a commutative algebra with unit A on a commutative field K of characteristic zero, is a r-Jacobi algebra,

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

such that

That ends the proof.

Canonical form Associated with a r-Jacobi AlgebraIn 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

The unique A-alternating multinear map of degree 2r

induce an unique A-linear map

such that

for all

We say that

Corollary 1 For all

for any

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

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