Full quivers of representations of algebras

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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 Zariski closed 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: Frobenius (along the diagonal) and, more generally, 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.

Original languageEnglish
Pages (from-to)5525-5569
Number of pages45
JournalTransactions of the American Mathematical Society
Volume364
Issue number10
DOIs
StatePublished - 2012

All Science Journal Classification (ASJC) codes

  • Applied Mathematics
  • General Mathematics

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