Necessary and sufficient conditions for a variety of Leibniz algebras to have polynomial growth

Abstract

We study the behavior of the codimension sequence of polynomial identities of Leibniz algebras over a field of characteristic 0. We prove that a variety V has polynomial growth if and only if the condition $$ N_2 A,\widetilde{V_1 } \not\subset V \subset \widetilde{N_c A} $$ holds, where N 2 A is the variety of Lie algebras defined by the identity \( (x_1 x_2 )(x_3 x_4 )(x_5 x_6 ) \equiv 0,\widetilde{V_1 } \) is the variety of Leibniz algebras defined by the identity x 1(x 2 x 3)(x 4 x 5) ≡ 0, and \( \widetilde{N_c A} \) is the variety of Leibniz algebras defined by the identity (x 1 x 2) … (x 2c+1 x 2c+2) ≡ 0.