A Lie Bracket on the Space of (Right) Biderivations of a Lie Algebra

TL;DR

This paper introduces a Lie bracket structure on the space of right biderivations of Lie algebras, exploring their algebraic properties and potential links to Lie groups, with implications for deformation theory and symmetries.

Contribution

It defines a Lie algebra structure on biderivations and initiates the construction of associated Lie groups, advancing the understanding of higher-order symmetries in Lie algebras.

Findings

Lie brackets on biderivations are established

Analysis of symmetric and skew-symmetric cases

Connection between infinitesimal and global structures

Abstract

Derivations extend the concept of differentiation from functions to algebraic structures as linear operators satisfying the Leibniz rule. In Lie algebras, derivations form a Lie algebra via the commutator bracket of linear endomorphisms. Motivated by this, we study biderivations-bilinear maps capturing higher-order infinitesimal symmetries. This work focuses on right biderivations of Lie algebras, introducing Lie brackets on spaces of biderivations to explore their algebraic and geometric properties. We analyse the interplay between left and right biderivation brackets through symmetric and skew-symmetric cases, providing a coherent Lie algebra framework. Moreover, we initiate the construction of Lie groups corresponding to the Lie algebra of biderivations, linking infinitesimal and global structures. Our results offer new perspectives on higher-order derivations with potential applications in deformation theory and generalised symmetry studies.