Pseudo-Riemannian Symmetries on Heisenberg group $mathbb{H}_{3}$

The notion of $Gamma$-symmetric space is a natural generalization of the classical notion of symmetric space based on $z_2$-grading of Lie algebras. In our case, we consider homogeneous spaces $G/H$ such that the Lie algebra $g$ of $G$ admits a $Gamma$-grading where $Gamma$ is a finite abelian group. In this work we study Riemannian metrics and Lorentzian metrics on the Heisenberg group $mathbb{H}_3$ adapted to the symmetries of a $Gamma$-symmetric structure on $mathbb{H}_3$. We prove that the classification of $z_2^2$-symmetric Riemannian and Lorentzian metrics on $mathbb{H}_3$ corresponds to the classification of left invariant Riemannian and Lorentzian metrics, up to isometries. This gives examples of non-symmetric Lorentzian homogeneous spaces.