PI-varieties associated to full quivers of representations of algebras

What is it about?

This is one of a series of papers by the authors that develop the theory needed to support Belov's proof of Specht's Conjecture for affine algebras over finite fields. More specifically, the authors note that the main motivation for this paper is to obtain a nonzero T-ideal of a representable algebra over an integral domain that has desirable properties. The existence of such T-ideals provides an inductive step for the proof of Specht's Conjecture mentioned above [A. Kanel-Belov, L. H. Rowen and U. Vishne, "Specht's problem for associative affine algebras over commutative Noetherian rings'', Trans. Amer. Math. Soc., in press]. The paper under review builds on the paper [A. Kanel-Belov, L. H. Rowen and U. Vishne, Trans. Amer. Math. Soc. 364 (2012), no. 10, 5525–5569; MR2931338], where a full quiver and a pseudo-quiver of a representation were defined. A full quiver is a cycle-free directed graph endowed with additional information encoded in the "glueing'' of vertices and of edges. A full quiver is also a covering of the classical quiver and the additional information in a full quiver helps one analyze Zariski closed algebras over finite fields as well as over other non-algebraically closed fields. A pseudo-quiver is similar but allows for a linear change of basis in the representation. In the current paper, the authors study how one can pass between full quivers and varieties of algebras. They then use the full quiver of an algebra to obtain PIs with the desired properties as well as build PI-algebras from full quivers. The main theorem of this paper is the Trace Adjunction Theorem, which states that for any maximal subpath of a basic full quiver of a representable relatively free algebra $A$ over a finite field, there is a naturally described polynomial whose T-ideal is also an ideal of the algebra $\widehat{A}$ obtained by taking $A$ together with the traces adjoined as per Shirshov's Theorem. (A full quiver is called basic if it has a unique initial vertex and a unique terminal vertex.) The Trace Adjunction Theorem is an inductive tool which is a key feature in the proof of Specht's problem over finite fields mentioned above. Consequences of this theorem discussed in this paper include a straightforward proof of the rationality of the Hilbert series of a relatively free PI-algebra (outlined in §6) and the construction of a parametric PI which is not defined over the base field (given in Example 7.2). Reviewed by George F. Seelinger

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