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

S. P. Mishchenko ; O. I. Cherevatenko

Fundamentalʹnaâ i prikladnaâ matematika, Tome 12 (2006) no. 8, pp. 207-215

Résumé

We study the behaviour of the codimension sequence of polynomial identities of Leibniz algebras over a field of characteristic 0. We prove that a variety $\mathbf V$ has polynomial growth if and only if the condition $$ \mathbf N_2\mathbf A,\widetilde{\mathbf V_1}\not\subset\mathbf V\subset\widetilde{\mathbf N_c\mathbf A} $$ holds, where $\mathbf N_2\mathbf A$ is the variety of Lie algebras defined by the identity $$ (x_1x_2)(x_3x_4)(x_5x_6)\equiv 0, $$ $\widetilde{\mathbf V_1}$ is the variety of Leibniz algebras defined by the identity $$ x_1(x_2x_3)(x_4x_5)\equiv 0, $$ and $\widetilde{\mathbf N_c\mathbf A}$ is the variety of Leibniz algebras defined by the identity $$ (x_1x_2)\cdots(x_{2c+1}x_{2c+2})\equiv 0. $$