On the Existence and Properties of Left Invariant $k$-Symplectic Structures on Lie Groups with Bi-Invariant Pseudo-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 $\mathrm{SL}(n,\mathbb{R})$. 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 $\mathrm{SL}(2, \mathbb{R})$ with a 2-symplectic structure, the universal covering of the Lorentz group $\mathrm{SO}(3, 1)$ 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.