The graded preprojective algebra of a quiver

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Abstract

The paper introduces a new grading on the preprojective algebra of an arbitrary locally finite quiver. Viewing the algebra as a left module over the path algebra, the author uses the grading to give an explicit geometric construction of a canonical collection of exact sequences of its submodules. If a vertex of the quiver is a source, the above submodules behave nicely with respect to the corresponding reflection functor. It follows that when the quiver is finite and without oriented cycles, the canonical exact sequences are the almost split sequences with preprojective terms, and the indecomposable direct summands of the submodules are the non-isomorphic indecomposable preprojective modules. The proof extends that given by Gelfand and Ponomarev in the case when the finite quiver is a tree.

Original languageEnglish (US)
Pages (from-to)13-22
Number of pages10
JournalBulletin of the London Mathematical Society
Volume36
Issue number1
DOIs
StatePublished - Jan 2004

ASJC Scopus subject areas

  • Mathematics(all)

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