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.

References in corpus (1)

• Weakly associative algebras, Poisson algebras and quantization

Cited by in corpus (11)

• Transposed Poisson structures on Block Lie algebras and superalgebras
• Transposed Poisson structures on Witt type algebras
• Transposed Poisson structures on Galilean and solvable Lie algebras
• The algebraic and geometric classification of nilpotent weakly associative and symmetric Leibniz algebras
• Transposed Poisson structures on generalized Witt algebras and Block Lie algebras
• Transposed Poisson structures
• Transposed Poisson structures on solvable and perfect Lie algebras
• Quasi-triangular, factorizable Leibniz bialgebras and relative Rota-Baxter operators
• Metrics related to Oscillator algebras
• $\frac{1}{2}$-derivations of Lie algebras and transposed Poisson algebras
• Skew-adjoint maps and quadratic Lie algebras