Representations, extensions and deformations of $n$-BiHom-Lie algebras

TL;DR

This paper explores the structure and deformation theory of $n$-BiHom-Lie algebras, introducing new extensions, characterizing quadratic cases, and linking cohomology to deformation properties.

Contribution

It defines representations, introduces $T_{ heta}$-extensions, characterizes quadratic $n$-BiHom-Lie algebras, and develops deformation theory using cohomology.

Findings

Characterization of $T_{ heta}^{ ext{*}}$-extensions for quadratic $n$-BiHom-Lie algebras

Development of deformation theory via first and second cohomology groups

Conditions for isomorphism to $T_{ heta}^{ ext{*}}$-extensions

Abstract

In this paper we define and discuss the representations of $n$-BiHom-Lie algebra. We also introduce $T_{\theta}$-extensions and $T_{\theta}^{\ast}$-extensions of $n$-BiHom-Lie algebras and prove the necessary and sufficient conditions for a $2m$-dimensional quadratic $n$-Bihom-Lie algebra to be isomorphic to a $T_{\theta}^{\ast}$-extension. Moreover, we develop the one-parameter formal deformations of $n$-BiHom-Lie algebras, and we proved that the first and second cohomology groups are suitable to the deformation theory involving infinitesimals, equivalent deformations, and rigidity