Symmetric Biderivations on Complex Semisimple Lie Algebras

TL;DR

This paper proves that all symmetric biderivations on finite-dimensional complex semisimple Lie algebras are trivial, and explores their applications to various algebraic structures including Takiff and symplectic oscillator algebras.

Contribution

It establishes the triviality of symmetric biderivations on complex semisimple Lie algebras and characterizes biderivations on related algebraic structures.

Findings

All symmetric biderivations on complex semisimple Lie algebras are trivial.

Explicit descriptions of biderivations on Takiff and symplectic oscillator algebras.

Classification of supersymmetric biderivations on certain Lie superalgebras.

Abstract

This paper studies biderivations on finite-dimensional complex semisimple Lie algebras to their finite-dimensional modules. More precisely, we prove that all such symmetric biderivations are trivial. As applications, we determine all biderivations on Takiff algebras and symplectic oscillator algebras, as well as all supersymmetric biderivations over some Lie superalgebras including all finite-dimensional classical simple Lie superalgebras.