On minimal Leibniz algebras with nilpotent commutator subalgebra

Abstract:

Let $\{c_n({\mathbf V})\}_{n\geq 1}$ be the codimension sequence of a variety of Leibniz algebras ${\mathbf V}$. The complexity function $\mathcal {C}({\mathbf V},z)=\sum _{n=1}^{\infty }c_n({\mathbf V})z^n/n!$ is studied. This is an exponential generating function for the codimension sequence. Before, complexity functions were used to study Lie algebras and associative algebras. In this paper, an explicit formula is obtained for the complexity function of a variety of Leibniz algebras with nilpotent commutator subalgebra, specifically, the variety determined by the identity $x_0(x_1x_2)(x_3x_4)\dots (x_{2s-1}x_{2s})=0$. By using this function, an explicit formula is derived for the codimensions of these algebras, which grow exponentially. Also, two series of varieties of Leibniz algebras with nilpotent commutator subalgebra of polynomial growth are constructed; they are minimal in a certain sense. Namely, the codimension sequence of any variety in the first of these series grows as a polynomial of a certain degree $k$, but for all its proper subvarieties the codimension sequence grows as a polynomial of some degree strictly smaller than $k$. The codimension sequence of any variety of the second series grows as a polynomial with some value of the leading coefficient $q$, whereas for all its proper subvarieties the codimension sequence grows as a polynomial whose leading coefficient is strictly smaller than $q$.