Hagen Meltzer

Publications (6)0.6 Total impact

• Article: Triangle singularities, ADE-chains, and weighted projective lines

  Dirk Kussin, Helmut Lenzing, Hagen Meltzer
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  ABSTRACT: We investigate the triangle singularity $f=x^a+y^b+z^c$, or $S=k[x,y,z]/(f)$, attached to a weighted projective line $X$ given by the weight triple $(a,b,c)$. We investigate the stable category of vector bundles on $X$ obtained from the vector bundles by factoring out all line bundles. This category is triangulated and has Serre duality. It is, moreover, naturally equivalent to the stable category of graded maximal Cohen-Macaulay modules over $S$ (or matrix factorizations of $f$), and then by results of Buchweitz and Orlov to the graded singularity category of $f$. We show that this category is fractional Calabi-Yau with a CY-dimension that is a function of the Euler characteristic of $X$. We show the existence of a tilting object which has the shape of an $(a-1)(b-1)(c-1)$-cuboid. Particular attention is given to the weight types $(2,a,b)$, yielding an explanation of Happel-Seidel symmetry for a class of important Nakayama algebras. In particular, the weight sequence $(2,3,p)$ corresponds to an ADE-chain, the $E_n$-chain, extrapolating the exceptional Dynkin cases $E_6$, $E_7$ and $E_8$ to a whole sequence of triangulated categories.
  03/2012;
• Source

  Available from: ArXiv

  Article: Nilpotent operators and weighted projective lines

  Dirk Kussin, Helmut Lenzing, Hagen Meltzer
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  ABSTRACT: We show a surprising link between singularity theory and the invariant subspace problem of nilpotent operators as recently studied by C. M. Ringel and M. Schmidmeier, a problem with a longstanding history going back to G. Birkhoff. The link is established via weighted projective lines and (stable) categories of vector bundles on those. The setup yields a new approach to attack the subspace problem. In particular, we deduce the main results of Ringel and Schmidmeier for nilpotency degree p from properties of the category of vector bundles on the weighted projective line of weight type (2,3,p), obtained by Serre construction from the triangle singularity x^2+y^3+z^p. For p=6 the Ringel-Schmidmeier classification is thus covered by the classification of vector bundles for tubular type (2,3,6), and then is closely related to Atiyah's classification of vector bundles on a smooth elliptic curve. Returning to the general case, we establish that the stable categories associated to vector bundles or invariant subspaces of nilpotent operators may be naturally identified as triangulated categories. They satisfy Serre duality and also have tilting objects whose endomorphism rings play a role in singularity theory. In fact, we thus obtain a whole sequence of triangulated (fractional) Calabi-Yau categories, indexed by p, which naturally form an ADE-chain.
  02/2010;
• Source

  Available from: ArXiv

  Article: Indecomposable representations for extended Dynkin quivers

  Dirk Kussin, Hagen Meltzer
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  ABSTRACT: We describe a method for an explicit determination of indecomposable preprojective and preinjective representations for extended Dynkin quivers by vector spaces and matrices. This method uses tilting theory and the explicit knowledge of indecomposable modules over the corresponding canonical algebra of domestic type.
  01/2007;
• Article: Exceptional Sequences Determined by their Cartan Matrix

  Helmut Lenzing, Hagen Meltzer
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  ABSTRACT: We investigate complete exceptional sequences E=(E 1,,E n ) in the derived category D b of finite-dimensional modules over a canonical algebra, equivalently in the derived category D b X of coherent sheaves on a weighted projective line, and the associated Cartan matrices C(E)=( [E i ],[E j ]). As a consequence of the transitivity of the braid group action on such sequences we show that a given Cartan matrix has at most finitely many realizations by an exceptional sequence E, up to an automorphism and a multi-translation (E 1,,E n )(E 1[i 1],,E n [i n ]) of D b . Moreover, we determine a bound on the number of such realizations. Our results imply that a derived canonical algebra A is determined by its Cartan matrix up to isomorphism if and only if the Hochschild cohomology of A vanishes in nonzero degree, a condition satisfied if A is representation-finite.
  Algebras and Representation 01/2002; 5(2):201-209. · 0.60 Impact Factor
• Article: The automorphism group of the derived category for a weighted projective line

  Helmut Lenzing, Hagen Meltzer
  Communications in Algebra. 01/2000; 28(4):1685-1700.
• Article: Indecomposable modules for domestic canonical algebras

  Dirk Kussin, Hagen Meltzer
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  ABSTRACT: We show that–up to precisely one–each exceptional module over a domestic canonical algebra of quiver type over a field k can be represented by matrices whose entries are just 0 and 1. In the case we calculate the matrices of these representations explicitly.
  Journal of Pure and Applied Algebra.