Minimal varieties of associative algebras and transcendental series

A variety of associative algebras over a field of characteristic 0 is called minimal if the exponent of the variety which measures the growth of its codimension sequence is strictly larger than the exponent of any of its proper subvarieties, i.e., its codimension sequence grows much faster than the codimension sequence of its proper subvarieties. By the results of Giambruno and Zaicev it follows that the number [Formula: see text] of minimal varieties of given exponent [Formula: see text] is finite. Using methods of the theory of colored (or weighted) compositions of integers, we show that the limit [Formula: see text] exists and can be expressed as the positive solution of an equation [Formula: see text] where [Formula: see text] is an explicitly given power series. Similar results are obtained for the number of minimal varieties with a given Gelfand–Kirillov dimension of their relatively free algebras of rank [Formula: see text]. It follows from classical results on lacunary power series that the generating function of the sequence [Formula: see text], [Formula: see text], is transcendental. With the same approach we construct examples of free graded semigroups [Formula: see text] with the following property. If [Formula: see text] is the number of elements of degree [Formula: see text] of [Formula: see text], then the limit [Formula: see text] exists and is transcendental.

ON VARIETIES OF ASSOCIATIVE ALGEBRAS WITH WEAK GROWTH

We prove that any variety of associative algebras with weak growth of the sequence {c_n(V)}_{n\geq 1} satisfies the identity [x_1, x_2][x_3, x_4] . . . [x_2_{s-1}, x_{2s}] = 0 for some s. As a consequence, the exponent of an arbitrary associative variety with weak growth exists and is an integer and if the characteristic of the ground field is distinct from 2 then there exists no varieties of associative algebras whose growth is intermediate between polynomial and exponential.

Classifying the Minimal Varieties of Polynomial Growth

Let ν be a variety of associative algebras generated by an algebra with 1 over a field of characteristic zero. This paper is devoted to the classification of the varieties ν that are minimal of polynomial growth (i.e., their sequence of codimensions grows like nk, but any proper subvariety grows like nt with t < k). These varieties are the building blocks of general varieties of polynomial growth.It turns out that for k ≤ 4 there are only a finite number of varieties of polynomial growth nk, but for each k > 4, the number of minimal varieties is at least |F|, the cardinality of the base field, and we give a recipe for their construction.

AUTOMORPHISMS OF THE ENDOMORPHISM SEMIGROUP OF A FREE ASSOCIATIVE ALGEBRA

Let [Formula: see text] be the variety of associative algebras over a field K and A = K 〈x1,…, xn〉 be a free associative algebra in the variety [Formula: see text] freely generated by a set X = {x1,…, xn}, End A the semigroup of endomorphisms of A, and Aut End A the group of automorphisms of the semigroup End A. We prove that the group Aut End A is generated by semi-inner and mirror automorphisms of End A. A similar result is obtained for the automorphism group Aut [Formula: see text], where [Formula: see text] is the subcategory of finitely generated free algebras of the variety [Formula: see text]. The later result solves Problem 3.9 formulated in [17].