Non-existence of symmetric biderivations on finite-dimensional perfect Lie algebras

TL;DR

This work resolves the existence question for symmetric biderivations on finite-dimensional perfect Lie algebras over characteristic zero by linking such biderivations to $A_{BD}$-structures. The authors first establish an equivalence between nontrivial symmetric biderivations and nonzero $A_{BD}$-structures, then develop a general framework for $A_{BD}$-structures via a Levi decomposition and associated maps $F$, $G$, and $\Delta$. They prove a reduction step showing any nonexistence result for the general case would follow from the abelian-radical case, and finally demonstrate that abelian-radical perfect Lie algebras admit only trivial $A_{BD}$-structures, with a detailed treatment of the $\mathfrak{sl}(2)$-Levi scenarios. The conclusion is that all symmetric biderivations into finite-dimensional modules vanish, answering Brešar and Zhao’s open question and implying that CPA-structures on such algebras are trivial as well. The result strengthens the understanding of derivation-like structures in Lie theory and rules out nontrivial $A_{BD}$-structures in the finite-dimensional perfect setting over characteristic zero.

Abstract

We show that there are no symmetric non-zero biderivations on perfect Lie algebras of finite dimension over a field of characteristic zero. We show that this is equivalent to show that every symmetric biderivation on a finite-dimensional perfect Lie algebra over such a field with values in a finite-dimensional module vanishes identically. This answers an open question posed by M. Brešar and K. Zhao.

Theorems & Definitions (35)

• Definition 2.1
• Definition 2.2
• Theorem 2.1
• proof
• Lemma 3.1
• Lemma 3.2
• Remark 3.1
• Proposition 3.1
• proof
• Corollary 3.1