On free products in varieties of associative algebras
type: Item
instance of: scholarly article
title: On free products in varieties of associative algebras
zbMATH Open document ID: 0953.16023
DOI: 10.1007/BF02743168
author: D. I. Piontkovskii
published in: Mathematical Notes
publication date: 2000-10-31T00:00:00Z
review text: The author studies the Hilbert series of relatively free algebras in varieties of associative algebras over a field of characteristic 0. \textit{A.~Ya.~Belov} proved that in any such variety \(\mathcal V\) the Hilbert series of the \(\mathcal V\)-free object of countable rank is rational [see Usp. Mat. Nauk 52, No. 2, 153-154 (1997; Zbl 0919.16018)]. Let \(Vect\), \(Com\) and \(Ass\) be, respectively, the variety of all algebras with zero multiplication, the variety of all commutative algebras and the variety of all associative ones. It is known that for these three varieties a stronger assertion than the one cited above holds. Namely, for these three varieties there are well known formulas that rationaly express the Hilbert series of a free product of algebras via the Hilbert series of the factors. The main result of the article is that there is no formula of such a kind in any variety excluding the three ones mentioned. Another result is that there is a similar formula for infinitely many varieties in the partial case when we restrict ourselves to the class of graded PI-algebras generated by their components of degree one.
Mathematics Subject Classification ID: 16R10
Mathematics Subject Classification ID: 16S10
Mathematics Subject Classification ID: 16W50
zbMATH DE Number: 1439577
zbMATH Keywords: relatively free algebras
zbMATH Keywords: varieties of associative algebras
zbMATH Keywords: Hilbert series
zbMATH Keywords: free products
zbMATH Keywords: graded algebras
zbMATH Keywords: PI-algebras
reviewed by: Boris Vernikov
MaRDI profile type: Publication
cites work: On the rationality of Hilbert series of relatively free algebras
cites work: Q4129587
cites work: Q3487576
full work available at URL: https://doi.org/10.1007/bf02743168
OpenAlex ID: W2029242866
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description: scientific article; zbMATH DE number 1439577