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

Let m, n be integers with m>n. Denote Mm,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∈Mm,n. Then Mm,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.