Lie Biderivations on Triangular Algebras

TL;DR

This paper characterizes Lie biderivations on triangular algebras, showing they decompose into sums of specific types of biderivations and central mappings, with applications to block upper triangular and Hilbert space nest algebras.

Contribution

It provides a description of Lie biderivations on triangular algebras, including their decomposition into inner, extremal, and central parts, under mild assumptions.

Findings

Lie biderivations decompose into inner, extremal, and central components.

Results apply to block upper triangular and Hilbert space nest algebras.

Provides a structural understanding of Lie biderivations in these algebraic settings.

Abstract

Let $\mathcal{T}$ be a triangular algebra over a commutative ring $\mathcal{R}$ and $\varphi: \mathcal{T} \times \mathcal{T}\longrightarrow \mathcal{T}$ be an arbitrary Lie biderivation of $\mathcal{T}$. We will address the question of describing the form of $\varphi$ in the current work. It is shown that under certain mild assumptions, $\varphi$ is the sum of an inner biderivation and an extremal biderivation and a some central bilinear mapping. Our results is immediately applied to block upper triangular algebras and Hilbert space nest algebras .