Biderivations of complete Leibniz algebras
Alfonso Di Bartolo, Francesco Paolo Di Fatta, Gianmarco La Rosa
TL;DR
This work addresses how biderivations behave on complete Leibniz algebras under two competing definitions of completeness. It characterizes biderivations by expressing them through linear maps $\varphi,\psi$ (and Leibniz-kernel corrections for def1) so that $f(x,y)=[\varphi(x),y]$ and related forms capture the bilinear map's action in each argument. It also analyzes symmetric and skew-symmetric biderivations via commuting and skew-commuting maps, and clarifies when such a representation is possible under each completeness notion. The results extend known Lie-algebra completeness cases to the Leibniz setting and highlight structural differences between the two definitions, with implications for the understanding of inner derivations and central structures in Leibniz algebras.
Abstract
If one wishes to define a complete Leibniz algebra in such a way as to extend the notion of a complete Lie algebra, two distinct definitions can be found in the current literature. Since biderivations on complete Lie algebras have already been studied, in order to extend those results and considering that Leibniz algebras are, among others, a natural generalisation of Lie algebras, we study here the biderivations of complete Leibniz algebras according to both definitions. In each case, we provide necessary and sufficient conditions for a bilinear map to be a biderivation of a Leibniz algebra. Finally, we analyse both symmetric and skew-symmetric biderivations, highlighting their structural properties.