On almost nilpotent varieties of anticommutative metabelian algebras

Let $\Phi$ be a field of characteristic zero. We consider variety of anticommutative metabelian algebras, denoted $\mathbf{MA}$, in which the anticommutativity identity $x_1x_2\equiv-x_2x_1$ and the metabelian identity $(x_1x_2)(x_3x_4)\equiv0$ are satisfied. The associativity of multiplication is not assumed. Numerical invariants of the variety of all anticommutative metabelian algebras are obtained: the sequence of codimensions is $c_n(\mathbf{MA})={n!}/2$. An algorithm for computing the multiplicities of $m_\lambda(\mathbf{MA})$ for $n>2$ is presented. We define a series of anticommutative metabelian algebras $C_m$ for any integer $m\ge2$ and prove the existence of almost nilpotent variety with PI-exponent of $m$. Moreover, two almost nilpotent varieties of subexponential growth are studied. The first variety is the well-known variety of all metabelian Lie algebras, denoted $\mathbf A^2 $, the second – the almost nilpotent variety $\mathbf V_\mathrm{anti}$ generated by the anticommutative metabelian algebra $G$, $\mathbf V_\mathrm{anti}=\operatorname{var}(G)$, which is defined in our investigation. In case of varieties of anticommutative metabelian algebras, it is shown that there are only two almost nilpotent varieties of subexponential growth: $\mathbf A^2$ and $\mathbf V_\mathrm{anti}$. The proofs are based on the theory of irreducible modules, Young diagram and tableau, and some basic notions of the representation theory for the symmetric group. All results are obtained by means of combinatorial methods.