Received 13 November 2015; accepted 7 March 2016; published 10 March 2016
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
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
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.
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
Thus, the existence of an unique A-linear map
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
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
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
We say that is the canonical form associated with the r-Jacobi algebra A.
Corollary 1 For all
The author thanks Prof. E. Okassa for his remarks and sugestions.