Poisson algebras and symmetric Leibniz bialgebra structures on oscillator Lie algebras

Abstract

Oscillator Lie algebras are the only non commutative solvable Lie algebras which carry a bi-invariant Lorentzian metric. In this paper, we determine all the Poisson structures, and in particular, all symmetric Leibniz algebra structures whose underlying Lie algebra is an oscillator Lie algebra. We give also all the symmetric Leibniz bialgebra structures whose underlying Lie bialgebra structure is a Lie bialgebra structure on an oscillator Lie algebra. We derive some geometric consequences on oscillator Lie groups.

Weakly associative and symmetric Leibniz algebras

“…We find again the result proved in [1] in which one shows that the oscillator Lie algebra can be endowed with a symmetric Leibniz algebra structure and with a Poisson algebra structure.…”

“…So a Poisson algebra can be naturally obtained from each symmetric Leibniz algebra by polarization. We find a result of [1].…”

“…We use this property to describe these algebras which leads to a structure theorem. As an application we find in particular some results of [1].…”