Complexity Functions of Varieties of Leibniz Algebras with Nilpotent Commutator Subalgebra

摘要

The Leibniz algebras are defined by the identity (xy)z = (xz)y + x(yz); they are generalizations of Lie algebras. Leibniz algebras are intensively studied since the nineties, beginning with the paper of Loday. These algebras are naturally related to differential geometry, classical algebraic topology, and noncommutative geometry. Let L(X) be a free Leibniz algebra over a field K, where X = {x_1, x_2, … } is a countable set of free generators, and let P_n be the subspace of L(X) consisting of all multilinear elements of degree n in the variables xi,…,xn. Let Ⅴ be a variety of Leibniz algebras, and let Id(Ⅴ) be the ideal of identities of the variety Ⅴ in the free algebra L(X) (for the necessary definitions and information concerning the theory of Pl-algebras, see, e.g., the monograph). Write P_n(Ⅴ) = P_n/{P_n ∩ Id(Ⅴ)), c_n(Ⅴ) = dim P_n(Ⅴ). We omit the brackets under their left-normed arrangement, i.e., ((ab)c) = abc. In the next lemma, which can readily be verified, we present a construction of Leibniz algebras using associative algebras.