Details
| Original language | English |
|---|---|
| Pages (from-to) | 25-47 |
| Number of pages | 23 |
| Journal | Pacific journal of mathematics |
| Volume | 229 |
| Issue number | 1 |
| Publication status | Published - 1 Jan 2007 |
Abstract
Cartan matrices are of fundamental importance in representation theory. For algebras defined by quivers with monomial relations, the computation of the entries of the Cartan matrix amounts to counting nonzero paths in the quivers, leading naturally to a combinatorial setting. For derived module categories, the invariant factors, and hence the determinant, of the Cartan matrix are preserved by derived equivalences. In the generalization called q-Cartan matrices (the classical Cartan matrix corresponding to q = 1), each nonzero path is weighted by a power of an indeterminate q according to its length. We study q-Cartan matrices for gentle and skewed-gentle algebras, which occur naturally in representation theory, especially in the context of derived categories. We determine normal forms for these matrices in the skewed-gentle case, giving explicit combinatorial formulae for the invariant factors and the determinant. As an application, we show how to use our formulae for the difficult problem of distinguishing derived equivalence classes.
Keywords
- Cartan matrices, Derived categories, Skewed-gentle algebras
ASJC Scopus subject areas
- Mathematics(all)
- General Mathematics
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In: Pacific journal of mathematics, Vol. 229, No. 1, 01.01.2007, p. 25-47.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - q-Cartan matrices and combinatorial invariants of derived categories for skewed-gentle algebras
AU - Bessenrodt, Christine
AU - Holm, Thorsten
PY - 2007/1/1
Y1 - 2007/1/1
N2 - Cartan matrices are of fundamental importance in representation theory. For algebras defined by quivers with monomial relations, the computation of the entries of the Cartan matrix amounts to counting nonzero paths in the quivers, leading naturally to a combinatorial setting. For derived module categories, the invariant factors, and hence the determinant, of the Cartan matrix are preserved by derived equivalences. In the generalization called q-Cartan matrices (the classical Cartan matrix corresponding to q = 1), each nonzero path is weighted by a power of an indeterminate q according to its length. We study q-Cartan matrices for gentle and skewed-gentle algebras, which occur naturally in representation theory, especially in the context of derived categories. We determine normal forms for these matrices in the skewed-gentle case, giving explicit combinatorial formulae for the invariant factors and the determinant. As an application, we show how to use our formulae for the difficult problem of distinguishing derived equivalence classes.
AB - Cartan matrices are of fundamental importance in representation theory. For algebras defined by quivers with monomial relations, the computation of the entries of the Cartan matrix amounts to counting nonzero paths in the quivers, leading naturally to a combinatorial setting. For derived module categories, the invariant factors, and hence the determinant, of the Cartan matrix are preserved by derived equivalences. In the generalization called q-Cartan matrices (the classical Cartan matrix corresponding to q = 1), each nonzero path is weighted by a power of an indeterminate q according to its length. We study q-Cartan matrices for gentle and skewed-gentle algebras, which occur naturally in representation theory, especially in the context of derived categories. We determine normal forms for these matrices in the skewed-gentle case, giving explicit combinatorial formulae for the invariant factors and the determinant. As an application, we show how to use our formulae for the difficult problem of distinguishing derived equivalence classes.
KW - Cartan matrices
KW - Derived categories
KW - Skewed-gentle algebras
UR - http://www.scopus.com/inward/record.url?scp=34548428159&partnerID=8YFLogxK
U2 - 10.2140/pjm.2007.229.25
DO - 10.2140/pjm.2007.229.25
M3 - Article
AN - SCOPUS:34548428159
VL - 229
SP - 25
EP - 47
JO - Pacific journal of mathematics
JF - Pacific journal of mathematics
SN - 0030-8730
IS - 1
ER -