Symplectic Leibniz algebras as a non-commutative version of symplectic Lie algebras

TL;DR

This paper introduces symplectic Leibniz algebras as a non-commutative generalization of symplectic Lie algebras, providing new constructions and classifications for these algebraic structures.

Contribution

It defines symplectic Leibniz algebras, explores their properties, and presents a method to construct them from symplectic Lie algebras using double extension and T*-extension techniques.

Findings

Symplectic Leibniz algebras generalize symplectic Lie algebras.

Complete classification of symmetric Leibniz algebras that are symplectic.

Construction method from symplectic Lie algebras and vector spaces.

Abstract

We introduce symplectic left Leibniz algebras and symplectic right Leibniz algebras as generalizations of symplectic Lie algebras. These algebras possess a left symmetric product and are Lie-admissible. We describe completely symmetric Leibniz algebras that are symplectic as both left and right Leibniz algebras. Additionally, we show that symplectic left or right Leibniz algebras can be constructed from a symplectic Lie algebra and a vector space through a method that combines the double extension process and the $T^*$-extension. This approach allows us to generate a broad class of examples.