On almost nilpotent varieties of anticommutative metabelian algebras | Prikladnaya Diskretnaya Matematika - Applied Discrete Mathematics. 2017. № 38. DOI: 10.17223/20710410/38/2

Let Ф be a field of characteristic zero. We consider variety of anticommutative metabelian algebras, denoted MA, in which the anticommutativity identity x\x₂ = = -x₂xi and the metabelian identity (хіХ2)(хзХ4) = 0 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 Cn(MA) = n!/2. An algorithm for computing the multiplicities of тл(МА) for n > 2 is presented. We define a series of anticommutative metabelian algebras for any integer m @ 2 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 A², the second - the almost nilpotent variety V_anti generated by the anticommutative metabelian algebra G, V_anti = 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: A² and V_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.