Communications in Algebra ® , 39: 4536–4551, 2011 Copyright © Taylor & Francis Group, LLC ISSN: 0092-7872 print/1532-4125 online DOI: 10.1080/00927872.2011.616432 APPLICATION OF FULL QUIVERS OF REPRESENTATIONS OF ALGEBRAS, TO POLYNOMIAL IDENTITIES Alexei Belov-Kanel, Louis Rowen, and Uzi Vishne Department of Mathematics, Bar-Ilan University, Ramat-Gan, Israel In [7] we introduced the notion of full quivers of representations of algebras, which are more explicit than quivers of algebras, and better suited for algebras over finite fields. Here, we consider full quivers as a combinatorial tool in order to describe PI-varieties of algebras. We apply the theory to clarify the proofs of diverse topics in the literature: Determining which relatively free algebras are weakly Noetherian, determining when relatively free algebras are finitely presented, presenting a quick proof for the rationality of the Hilbert series of a relatively free PI-algebra, and explaining counterexamples to Specht’s conjecture for varieties of Lie algebras. Key Words: Full quiver; Hilbert series; Polynomial identities; Specht’s conjecture; Weakly Noetherian. 2000 Mathematics Subject Classification: Primary 16R10, 16R20, 16R40, 16G20; Secondary 16P40, 16P90. Dedicated to our friend and colleague Mia Cohen on the occasion of her retirement. 1. INTRODUCTION Recall [5, pp. 28ff.] that an algebra A (not necessarily with a unit element 1) over a field F is representable if it can be embedded as an F -subalgebra of M n K for a suitable field K ⊃ F . Throughout, the given base field F has order q (where q could be infinity), and K is taken to be algebraically closed. Note when q<  that q is a power of p = charF <  and the Frobenius map a → a q is an F -algebra endomorphism. We also assume that A is Zariski closed, i.e., closed in the Zariski topology of M n K. This is a generalization of finite dimensional algebras, studied in [6], which is particularly useful when the base field is finite. In [7] we consider the full quiver of a linear representation of an associative algebra, which is somewhat more explicit than the “classical” quiver of an algebra, and which enables one to study the structure of Zariski closed algebras. We determined properties of the full quivers by means of a close examination of the structure of Zariski closed algebras. Then the full quiver was used to study interactions between the radical and the semisimple component of Zariski closed algebras. Received December 15, 2010. Communicated by S. Montgomery. Address correspondence to Louis Rowen, Department of Mathematics, Bar-Ilan University, Ramat-Gan 52900, Israel; E-mail: rowen@macs.biu.ac.il 4536