Nilpotency of Zinbiel Algebras

Abstract

Zinbiel algebras are defined by the identity (a ? b) ? c = a?(b?c+c?b). We prove an analog of the Nagata--Higman theorem for Zinbiel algebras. We establish that every finite-dimensional Zinbiel algebra over an algebraically closed field is solvable. Every solvable Zinbiel algebra with solvability length N is a nil-algebra with nil-index 2N if p = char K = 0 or p > 2N ? 1. Conversely, every Zinbiel nil-algebra with nil-index N is solvable with solvability length N if p = 0 or p > N ? 1. Every finite-dimensional Zinbiel algebra over complex numbers is nilpotent, nil, and solvable.