Constructions and $T^{\ast}$-Extensions of 3-Bihom-Lie Superalgebras

TL;DR

This paper generalizes the construction of 3-Bihom-Lie superalgebras, explores their properties and extensions, and characterizes quadratic cases through isomorphisms to T*-extensions.

Contribution

It introduces new methods for constructing and analyzing 3-Bihom-Lie superalgebras and their extensions, including conditions for isomorphism to T*-extensions.

Findings

Properties like nilpotency and solvability can be lifted to T*-extensions.

Provides conditions for isomorphism to T*-extensions in quadratic cases.

Studies representations and various extensions of 3-Bihom-Lie superalgebras.

Abstract

The aim of this paper is to generalise the construction of $3$-Bihom-Lie superalgebras and we provide some properties can be lifted to its $T^{\ast}$-extensions such as nilpotency, solvability and decomposition. We study the representations, $T_{\theta}$-extensions and $T^{\ast}_{\theta}$-extension of $3$-Bihom-Lie superalgebras and prove the necessary and sufficient conditions for a $2n$-dimensional quadratic $3$-Bihom-Lie superalgebra to be isomorphic to a $T^{\ast}_{\theta}$-extension.