Full quivers of representations of algebras

TL;DR

This paper introduces the full quiver concept for algebra representations, capturing detailed properties and combinatorial aspects, especially for \\Zcd\\ algebras, and demonstrates that any representable algebra can be faithfully represented by such a quiver.

Contribution

It defines the full quiver of a representation, extending classical quivers to include vertex and arrow gluing, and shows all representable algebras can be described by these quivers with specific gluing types.

Findings

Full quivers capture subtle combinatorial properties of algebra representations.

Any representable algebra admits a faithful representation described by a full quiver with Frobenius or proportional Frobenius gluing.

Reductions related to polynomial identities are also discussed.

Abstract

We introduce the notion of the full quiver of a representation of an algebra, which is a cover of the (classical) quiver, but which captures properties of the representation itself. Gluing of vertices and of arrows enables one to study subtle combinatorial aspects of algebras which are lost in the classical quiver. Full quivers of representations apply especially well to \Zcd\ algebras, which have properties very like those of finite dimensional algebras over fields. By choosing the representation appropriately, one can restrict the gluing to two main types: {\it Frobenius} (along the diagonal) and, more generally {\it proportional} Frobenius gluing (above the diagonal), and our main result is that any representable algebra has a faithful representation described completely by such a full quiver. Further reductions are considered, which bear on the polynomial identities.