A New Approach to Leibniz Bialgebras

Abstract

International audienceA study of Leibniz bialgebras arising naturally through the double of Leibnizalgebras analogue to the classical Drinfeld’s double is presented. A key ingredient of ourwork is the fact that the underline vector space of a Leibniz algebra becomes a Lie algebraand also a commutative associative algebra, when provided with appropriate new products.A special class of them, the coboundary Leibniz bialgebras, gives us the natural frame-work for studying the Yang-Baxter equation (YBE) in our context, inspired in the classicalYang-Baxter equation as well as in the associative Yang-Baxter equation. Results of theexistence of coboundary Leibniz bialgebra on a symmetric Leibniz algebra under certainconditions are obtained. Some interesting examples of coboundary Leibniz bialgebras arealso included. The final part of the paper is dedicated to coboundary Leibniz bialgebrastructures on quadratic Leibniz algebras

Poisson algebras and symmetric Leibniz bialgebra structures on oscillator Lie algebras

“…Any Lie algebra is obviously a symmetric Leibniz algebra, however, the class of symmetric Leibniz algebras contains strictly the class of Lie algebras. In [2,Proposition 2.11], a useful characterization of symmetric Leibniz algebras is given. Note that symmetric Leibniz algebras constitute a subclass of left (right) Leibniz algebras introduced by Bloh [6,5] under the name of D-algebras and rediscovered by Loday [15].…”

“…As a Lie bialgebra is a Lie algebra g and a Lie bracket on its dual g * which are compatible in some sens (see Section 2), a symmetric Leibniz bialgebra is a symmetric Leibniz algebra A with a symmetric Leibniz product on its dual A * which are compatible. The notion of Lie bialgebras was introduced and studied by Drinfeld in [10] and symmetric Leibniz bialgebras were characterized by Barreiro and Benayadi in 2016 (see [2]). Note that any symmetric Leibniz bialgebra has an underlying structure of Lie bialgebra.…”

On Representations of Symmetric Leibniz Algebras

“…(A, •)). In [2], it was shown that (L, .) is a symmetric Leibniz algebra if and only if L − is a Lie algebra, L + is 3-nilpotent associative algebra (i.e.…”

“…is contained in the annihilator of the commutative associative algebra L + . The aim of this paper is to give a new approach to study the representations of the symmetric Leibniz algabras by using their characterization obtained in [2]. The idea is to use results on the representations of Lie algebras to obtain information on representations of symmetric Leibniz algebras.…”

“…is stable by π(x) and λ(x), ∀α ∈ L , ∀x ∈ L;(2) λ(x)(V α (L − )) = {0}, ∀α ∈ L \{0}, ∀x ∈ L; (3) if α ∈ L , there exists a basis of V α (L − ) such for any x ∈ L, the matrix A(x) := [a ij (x)] of π(x) | V α (L − ) with respect to this basis is upper triangular with a ii (x) = α(x), for all i; (4) there is a basis of V 0 (L − ) such for any x ∈ L, the matrices of π(x) | V 0 (L − ) and λ(x) | V0 (L − ) with respect to this basis is strictly upper triangular. Let (V, r, l) be a representation of a nilpotent symmetric Leibniz algebra (L, .)…”