Weighted locally gentle quivers and Cartan matrices |
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Authors: | Christine Bessenrodt Thorsten Holm |
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Affiliation: | a Institut für Algebra, Zahlentheorie und Diskrete Mathematik, Leibniz Universität Hannover, Welfengarten 1, D-30167 Hannover, Germany b Otto-von-Guericke-Universität Magdeburg, Institut für Algebra und Geometrie, Universitätsplatz 2, 39106 Madgeburg, Germany c Department of Pure Mathematics, University of Leeds, Leeds LS2 9JT, United Kingdom |
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Abstract: | We introduce and study the class of weighted locally gentle quivers. This naturally extends the class of gentle quivers and gentle algebras, which have been intensively studied in the representation theory of finite-dimensional algebras, to a wider class of potentially infinite-dimensional algebras. Weights on the arrows of these quivers lead to gradings on the corresponding algebras. For natural grading by path lengths, any locally gentle algebra is Koszul. The class of locally gentle algebras consists of the gentle algebras together with their Koszul duals.Our main result is a general combinatorial formula for the determinant of the weighted Cartan matrix of a weighted locally gentle quiver. We show that this weighted Cartan determinant is a rational function which is completely determined by the combinatorics of the quiver-more precisely by the number and the weight of certain oriented cycles. |
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Keywords: | Primary, 16G10, 16S37 secondary, 05E99, 11C20, 16G20, 18E30 |
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