| Abstract: | Varieties of associative algebras over a field of characteristic zero are considered. Belov recently proved that, in any variety
of this kind, the Hilbert series of a relatively free algebra of finite rank is rational. At the same time, for three important
varieties, namely, those of algebras with zero multiplication, of commutative algebras, and of all associative algebras, a
stronger assertion holds: for these varieties, formulas that rationally express the Hilbert series of the free product algebra
via the Hilbert series of the factors are well known. In the paper, a system of counterexamples is presented which shows that
there is no formula of this kind in any other variety, even in the case of two factors one of which is a free algebra. However,
if we restrict ourselves to the class of graded PI-algebras generated by their components of degree one, then there exist
infinitely many varieties for each of which a similar formula is valid.
Translated fromMatematicheskie Zametki, Vol. 65, No. 5, pp. 693–702, May, 1999. |
