Derivations and biderivations of the non-square linear Lie algebra gl(m×n, 𝔽)

Abstract

Let m, n be integers with $m>n$. Denote ${\mathbf{M}}_{m,n}$ by the set of all $m\times n$ matrices over the field $\mathbb{F}$ of characteristic zero. Let I be an $n\times m$ matrix with $(i,i)$-position 1 for any $1\le i\le n$, and 0 in other position. Define a bracket $[A,B]=\mathit{\text{AIB}}-\mathit{\text{BIA}}$, where $A,B\in {\mathbf{M}}_{m,n}$. Then ${\mathbf{M}}_{m,n}$ with this bracket is a Lie algebra, called non-square linear Lie algebra, denoted by $\text{gl}(m\times n,\mathbb{F})$. In this paper, all derivations and biderivations of $\text{gl}(m\times n,\mathbb{F})$ are determined. As applications of biderivations, the linear commuting maps and commutative post-Lie algebra structures on $\text{gl}(m\times n,\mathbb{F})$ are given.

Keywords:

• Biderivations
• derivations
• non-square linear Lie algebras

2020 Mathematics Subject Classification:

• 15B30
• 17B05
• 17B40

Additional information

Funding

This work was supported by Fujian Province Nature Science Foundation of China (2020J01364), Doctoral Research Launch Project of Longyan University (LB20202003), and Fujian Alliance of Mathematics (2023SXLMMS03).