On the existence and properties of left invariant k-symplectic structures on Lie groups with bi-invariant peudo-Riemannian metric

Abstract

k-symplectic manifolds are a convenient framework to study classical field theories and they are a generalization of polarized symplectic manifolds. This paper focus on the existence and the properties of left invariant k-symplectic structures on Lie groups having a bi-invariant pseudo-Riemannian metric. We show that compact semi-simple Lie groups and a large class of Lie groups having a bi-invariant pseudo-Riemannian metric does not carry any left invariant k-symplectic structure. This class contains the oscillator Lie groups which are the only solvable non abelian Lie groups having a bi-invariant Lorentzian metric. However, we built a natural left invariant n-symplectic structure on $\text{SL}(n,\mathbb{R})$. Moreover, up to dimension 6, only three connected and simply connected Lie groups have a bi-invariant indecomposable pseudo-Riemannian metric and a left invariant k-symplectic structure, namely, the universal covering of $\text{SL}(2,\mathbb{R})$ with a 2-symplectic structure, the universal covering of the Lorentz group $\text{SO}(3,1)$ with a 2-symplectic structure, and a 2-step nilpotent 6-dimensional connected and simply connected Lie group with both a 1-symplectic structure and a 2-symplectic structure.

KEYWORDS:

• Bi-invariant metric
• k-symplectic structure
• semi-simple Lie algebras

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 17B60
• 17B99