Biderivations of Null-Filiform and Filiform Zinbiel Algebras

Abstract

A biderivation of an algebra is a bilinear map which is a derivation in each argument. We describe all biderivations of finite-dimensional complex null-filiform and filiform Zinbiel algebras. For the n-dimensional null-filiform algebra, every biderivation is determined by an arbitrary value at its generating pair; the space of biderivations has dimension n and consists entirely of symmetric maps. For the three standard filiform families, we give explicit bases and determine their symmetric and skew-symmetric parts. We also treat all filiform algebras of dimensions at most four, including the exceptional families not covered by the classification in dimensions at least five. A three-dimensional algebra admits a three-dimensional space of skew-symmetric biderivations, two of which have values outside its annihilator. Up to isomorphism, it is the unique complex filiform Zinbiel algebra with this property. The proofs use generators, annihilator-valued maps, a truncated polynomial model, and conformal endomorphisms of bilinear forms.