TY - JOUR

T1 - The graded preprojective algebra of a quiver

AU - Kleiner, Mark

PY - 2004/1

Y1 - 2004/1

N2 - 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.

AB - 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.

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U2 - 10.1112/S0024609303002534

DO - 10.1112/S0024609303002534

M3 - Article

AN - SCOPUS:1442328305

SN - 0024-6093

VL - 36

SP - 13

EP - 22

JO - Bulletin of the London Mathematical Society

JF - Bulletin of the London Mathematical Society

IS - 1

ER -