PI-varieties associated to full quivers of representations of algebras. (Q2838079)

The paper under review is the forth paper in a series of papers by the same authors [Trans. Am. Math. Soc. 362, No. 9, 4695-4734 (2010; Zbl 1221.16021); Commun. Algebra 39, No. 12, 4536-4551 (2011; Zbl 1254.16016); Trans. Am. Math. Soc. 364, No. 10, 5525-5569 (2012; Zbl 1286.16019)]. In the previous papers the authors introduced the notion of the full quiver of a representation of an algebra. It is a cover of the (classical) quiver and allows to study combinatorial aspects of algebras which are lost in the classical quiver. Full quivers of representations work especially well for Zariski closed algebras and are better suited for algebras over finite fields.NEWLINENEWLINE In the paper under review the authors use full quivers and pseudo-quivers as a combinatorial tool in the study of varieties of algebras. Each full quiver is naturally associated to a polynomial that encapsulates trace-like properties of the underlying algebra. In several Canonization Theorems the authors show that full quivers of relatively free algebras can be reduced to full quivers of a better form. The constructions give interesting examples of polynomial identities (and non-identities).NEWLINENEWLINE As immediate applications of the methods and the results of the paper the authors give a straightforward proof of the rationality of the Hilbert series of relatively free algebras and construct parametric polynomial identities which are not defined over the base field. In a forthcoming paper the authors will use the results of the present paper for the solution of the Specht problem over finite fields.