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4 Colloquium Mathematicum

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Auslander-Reiten components for concealed-canonical algebras

100%

Meltzer H.

Colloquium Mathematicum

1996

tom 71

nr 2

183-202

Derived tubular algebras and APR-tilts

100%

Meltzer H.

nr 2

171-179

The global basie dimension of artin rings

71%

Meltzer H. , Skowroński A.

Instytut Matematyczny Polskiej Akademii Nauk

CONTENTS 1. Introduction.............5 2. Basic dimension of artin rings................7 3. Cobasic dimension of artin rings............8 4. Basic dimension of algebras stably equivalent to an hereditary artin algebra............12 5. Hereditary artin algebras of global basic and cobasic dimension 1....................17 6. Global basic and cobasic dimensions of radical squared zero algebras............34 References...............43

Indecomposable representations for extended Dynkin quivers of type 𝔼̃₈

63%

Kędzierski D. , Meltzer H.

Colloquium Mathematicum

2011

tom 124

nr 1

95-116

We discuss the problem of classification of indecomposable representations for extended Dynkin quivers of type 𝔼̃₈, with a fixed orientation. We describe a method for an explicit determination of all indecomposable preprojective and preinjective representations for those quivers over an arbitrary field and for all indecomposable representations in case the field is algebraically closed. This method uses tilting theory and results about indecomposable modules for a canonical algebra of type (5,3,2) obtained by Kussin and Meltzer and by Komoda and Meltzer. Using these techniques we calculate all series of preprojective indecomposable representations of rank 6. The same method has been used by Kussin and Meltzer to determine indecomposable representations for extended Dynkin quivers of type 𝔻̃ₙ and 𝔼̃₆. Moreover, our techniques can be applied to calculate indecomposable representations of extended Dynkin quivers of type 𝔼̃₇. The indecomposable representations for extended Dynkin quivers of type 𝔸̃ₙ are known.

Matrix factorizations for domestic triangle singularities

51%

Kędzierski D. , Lenzing H. , Meltzer H.

tom 140

nr 2

239-278

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}.