Details
| Original language | English |
|---|---|
| Pages (from-to) | 204-221 |
| Number of pages | 18 |
| Journal | Journal of Pure and Applied Algebra |
| Volume | 212 |
| Issue number | 1 |
| Early online date | 24 May 2007 |
| Publication status | Published - Jan 2008 |
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.
ASJC Scopus subject areas
- Mathematics(all)
- Algebra and Number Theory
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In: Journal of Pure and Applied Algebra, Vol. 212, No. 1, 01.2008, p. 204-221.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Weighted locally gentle quivers and Cartan matrices
AU - Bessenrodt, Christine
AU - Holm, Thorsten
PY - 2008/1
Y1 - 2008/1
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=34548441201&partnerID=8YFLogxK
U2 - 10.1016/j.jpaa.2007.05.004
DO - 10.1016/j.jpaa.2007.05.004
M3 - Article
AN - SCOPUS:34548441201
VL - 212
SP - 204
EP - 221
JO - Journal of Pure and Applied Algebra
JF - Journal of Pure and Applied Algebra
SN - 0022-4049
IS - 1
ER -