On $k$-para-Kähler Lie algebras a subclass of $k$-symplectic Lie algebras

Abstract

$k$-Para-Kähler Lie algebras are a generalization of para-Kähler Lie algebras $(k=1)$ and constitute a subclass of $k$-symplectic Lie algebras. In this paper, we show that the characterization of para-Kähler Lie algebras as left symmetric bialgebras can be generalized to $k$-para-Kähler 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-Kähler Lie algebras.

21 pages, 3 Tables

References in corpus (2)

• Left-symmetric Bialgebras and An Analogue of the Classical Yang-Baxter Equation
• Special symplectic Lie groups and hypersymplectic Lie groups