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

Abstract

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