Full quivers of representations of algebras

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.

44 pp

Cited by in corpus (5)

• Specht's problem for associative affine algebras over commutative Noetherian rings
• Identities of finitely generated graded algebras with involution
• Semisimple Algebras and PI-Invariants of Finite Dimensional Algebras
• Representability of relatively free affine algebras over a Noetherian ring
• Representability of affine algebras over an arbitrary field