On $k$-para-K\"ahler Lie algebras a subclass of $k$-symplectic Lie algebras

TL;DR

This paper extends the theory of para-K"ahler Lie algebras to the $k$-para-K"ahler case, introduces new algebraic structures, and classifies low-dimensional instances.

Contribution

It generalizes the characterization of para-K"ahler Lie algebras to the $k$-para-K"ahler setting and introduces two new structures that extend left symmetric algebras.

Findings

Classified all $(k+1)$-dimensional $k$-symplectic Lie algebras.

Determined all six-dimensional 2-para-K"ahler Lie algebras.

Introduced generalized $S$-matrices.

Abstract

$k$-Para-K\"ahler Lie algebras are a generalization of para-K\"ahler Lie algebras $(k=1)$ and constitute a subclass of $k$-symplectic Lie algebras. In this paper, we show that the characterization of para-K\"ahler Lie algebras as left symmetric bialgebras can be generalized to $k$-para-K\"ahler Lie algebras leading to the introduction of two new structures which are different but both generalize the notion of left symmetric algebra. This permits also the introduction of generalized $S$-matrices. We determine then all the $k$-symplectic Lie algebras of dimension $(k+1)$ and all the six dimensional 2-para-K\"ahler Lie algebras.