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1
Content available remote

On selfinjective algebras of Euclidean type

100%
Lenzing H., Skowroński A.
Colloquium Mathematicum
|
1999
|
tom 79
|
nr 1
71-76
2
Content available remote

Selfinjective algebras of wild canonical type

100%
Lenzing H., Skowroński A.
Colloquium Mathematicum
|
2003
|
tom 96
|
nr 2
245-275
EN
We develop the representation theory of selfinjective algebras which admit Galois coverings by the repetitive algebras of algebras whose derived category of bounded complexes of finite-dimensional modules is equivalent to the derived category of coherent sheaves on a weighted projective line with virtual genus greater than one.
3
Content available remote

Additive functions for quivers with relations

100%
Lenzing H., Reiten I.
Colloquium Mathematicum
|
1999
|
tom 82
|
nr 1
85-103
EN
Additive functions for quivers with relations extend the classical concept of additive functions for graphs. It is shown that the concept, recently introduced by T. Hübner in a special context, can be defined for different homological levels. The existence of such functions for level 2 resp. ∞ relates to a nonzero radical of the Tits resp. Euler form. We derive the existence of nonnegative additive functions from a family of stable tubes which stay tubes in the derived category, we investigate when this situation does appear and we study the restrictions imposed by the existence of a positive additive function.
4
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Quasi-tilted algebras of canonical type

100%
Lenzing H., Skowroński A.
Colloquium Mathematicum
|
1996
|
tom 71
|
nr 2
161-181
5
Content available remote

Roots of Nakayama and Auslander-Reiten translations

100%
Lenzing H., Skowroński A.
Colloquium Mathematicum
|
2000
|
tom 86
|
nr 2
209-230
EN
We discuss the roots of the Nakayama and Auslander-Reiten translations in the derived category of coherent sheaves over a weighted projective line. As an application we derive some new results on the structure of selfinjective algebras of canonical type.
6
Content available remote

Extremal properties for concealed-canonical algebras

81%
Barot M., Kussin D., Lenzing H.
Colloquium Mathematicum
|
2013
|
tom 130
|
nr 2
183-219
EN
Canonical algebras, introduced by C. M. Ringel in 1984, play an important role in the representation theory of finite-dimensional algebras. They also feature in many other mathematical areas like function theory, 3-manifolds, singularity theory, commutative algebra, algebraic geometry and mathematical physics. We show that canonical algebras are characterized by a number of interesting extremal properties (among concealed-canonical algebras, that is, the endomorphism rings of tilting bundles on a weighted projective line). We also investigate the corresponding class of algebras antipodal to canonical ones. Our study yields new insights into the nature of concealed-canonical algebras, and sheds a new light on an old question: Why are the canonical algebras canonical?
7
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Matrix factorizations for domestic triangle singularities

81%
Kędzierski D., Lenzing H., Meltzer H.
Colloquium Mathematicum
|
2015
|
tom 140
|
nr 2
239-278
EN
Working over an algebraically closed field k of any characteristic, we determine the matrix factorizations for the-suitably graded-triangle singularities $f = x^{a} + y^{b} + z^{c}$ of domestic type, that is, we assume that (a,b,c) are integers at least two satisfying 1/a + 1/b + 1/c > 1. Using work by Kussin-Lenzing-Meltzer, this is achieved by determining projective covers in the Frobenius category of vector bundles on the weighted projective line of weight type (a,b,c). Equivalently, in a representation-theoretic context, we can work in the mesh category of ℤ Δ̃ over k, where Δ̃ is the extended Dynkin diagram corresponding to the Dynkin diagram Δ = [a,b,c]. Our work is related to, but in methods and results different from, the determination of matrix factorizations for the ℤ-graded simple singularities by Kajiura-Saito-Takahashi. In particular, we obtain symmetric matrix factorizations whose entries are scalar multiples of monomials, with scalars taken from {0,±1}.
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