Left Nilpotent Research Articles

Combinatorics on Binary Words and Codimensions of Identities in Left Nilpotent Algebras

Numerical characteristics of polynomial identities of left nilpotent algebras are examined. Previously, we came up with a construction which, given an infinite binary word, allowed us to build a two-step left nilpotent algebra with specified properties of the codimension sequence. However, the class of the infinite words used was confined to periodic words and Sturm words. Here the previously proposed approach is generalized to a considerably more general case. It is proved that for any algebra constructed given a binary word with subexponential function of combinatorial complexity, there exists a PI-exponent, and its precise value is computed.

Some extremal properties of the variety of Leibniz algebras left nilpotent of class at most three

It is proved, for the case in which the ground field is of characteristic zero, that the variety of Leibniz algebras left nilpotent of class at most three is a variety of almost exponential growth with almost polynomial growth of the colength and has almost finite multiplicities.

Generating functions for ternary algebras and ternary trees

The main object of study are ternary algebras, i.e., algebras with a trilinear operation. In this class we study finitely generated algebras and their growth, as well as the growth of codimensions of absolutely free algebras and some other varieties. For these purposes we use ordinary generating functions and exponential generating functions (the complexity functions). In the classes of absolutely free, free symmetric, free antisymmetric, and some other algebras we study left nilpotent and completely left nilpotent algebras and varieties. The obtained results are equivalent to the enumeration of ternary trees which contain no forbidden subtrees of a special kind. As the main result, we prove that the complexity functions of the varieties of completely left nilpotent and left nilpotent ternary algebras are algebraic.

Automorphisms of Free Left Nilpotent Leibniz Algebras and Fixed Points#

ABSTRACT Let F m (N) be the free left nilpotent (of class two) Leibniz algebra of finite rank m, with m ≥ 2. We show that F m (N) has non-tame automorphisms and, for m ≥ 3, the automorphism group of F m (N) is generated by the tame automorphisms and one more non-tame IA-automorphism. Let F(N) be the free left nilpotent Leibniz algebra of rank greater than 1 and let G be an arbitrary non-trivial finite subgroup of the automorphism group of F(N). We prove that the fixed point subalgebra F(N) G is not finitely generated.

AN ORDER-THEORETIC PROPERTY OF THE COMMUTATOR

We describe a new order-theoretic property of the commutator for finite algebras. As a corollary we show that any right nilpotent congruence on a finite algebra is left nilpotent. The result is false for infinite algebras and the converse is false even for finite algebras. We show further that any solvable E-minimal algebra is left nilpotent, any finite algebra whose congruence lattice contains a 0, 1-sublattice isomorphic to M3 is left nilpotent and any homomorphic image of a finite abelian algebra is left and right nilpotent.