Full quivers of representations of algebras

What is it about?

A recent paper by the first author [Izv. Ross. Akad. Nauk Ser. Mat. 74 (2010), no. 1, 3–134; MR2655238] contains a bunch of important theorems that imply solutions to several longstanding problems in the theory of PI-rings. However, in the reviewer's opinion, arguments justifying these theorems in [op. cit.] can hardly be treated as proofs in the conventional meaning of the word. The authors characterize the present paper as a part of an ongoing project that has "to lay a firm foundation'' for one of the main results in [op. cit.]. Towards this aim, the authors introduce the notion of the full quiver of a representation of an algebra. The full quiver is a directed graph whose vertices are labeled in a special way to capture some properties of the representation. It is a cover of the classical quiver but has no directed cycles. The main result of the present paper is that any representable algebra has a faithful representation described completely by such a quiver. Reviewed by Mikhail V. Volkov

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