  $k$-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.
