Application of Full Quivers of Representations of Algebras, to Polynomial Identities

Abstract

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

ACKNOWLEDGMENTS

This research was supported by the Israel Science Foundation (grant No. 1178/06).

Notes

Communicated by S. Montgomery.

Dedicated to our friend and colleague Mia Cohen on the occasion of her retirement.