A New Definition of Superbiderivations for Lie Superalgebras

TL;DR

This paper introduces a novel definition of superbiderivations for Lie superalgebras, focusing on complete Lie superalgebras, and explores their structural, geometric, and deformation-theoretic implications.

Contribution

It proposes a new definition of superbiderivations that aligns with the Lie superalgebra bracket and studies their properties and applications in specific superalgebras.

Findings

New definition of superbiderivations for Lie superalgebras

Comparison with previous definitions and properties analyzed

Applications to superderivations of Heisenberg superalgebra and deformation theory

Abstract

In this paper, we study superbiderivations on Lie superalgebras from structural and geometric perspectives. Motivated by the classical fact that the bracket of a Lie algebra is itself a biderivation, we propose a new definition of superbiderivation for Lie superalgebras, one that requires the bracket to be a superbiderivation, a condition not satisfied by existing definitions in the literature. Our focus is on complete Lie superalgebras, a natural generalization of semisimple Lie algebras that has emerged as a promising framework in the search for alternative structural notions. In this setting, we introduce and study linear supercommuting maps, comparing our definition with previous proposals. Finally, we present two applications: one involving the superalgebra of superderivations of the Heisenberg Lie superalgebra and another offering initial geometric insights into deformation theory via superbiderivations.