Publications (12)2.6 Total impact
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ABSTRACT: We give a description of matrix bimodules parametrizing all indecomposable homogeneous Λmodules with a fixed integral slope over a tubular canonical algebra Λ, for all possible integers (Theorem 4.1). An important role in the first step of this description (Theorem 2.4) is played by the translation of the shift functor for coherent sheaves over the associated weighted projective line to the language of Λmodules (Theorem 3.2).Algebras and Representation 01/2014; · 0.55 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: We give an algorithmic description of matrix bimodules parametrizing all indecomposable homogeneous Λmodules with a fixed slope q over a tubular canonical algebra Λ, for all possible slopes q (Main Theorem 3.3). A crucial role in this description is played by universal extensions of bimodules and their nice properties (Theorems 3.1 and 3.2).Algebras and Representation 01/2014; · 0.55 Impact Factor  [Show abstract] [Hide abstract]
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 CohenMacaulay 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 CalabiYau with a CYdimension that is a function of the Euler characteristic of $X$. We show the existence of a tilting object which has the shape of an $(a1)(b1)(c1)$cuboid. Particular attention is given to the weight types $(2,a,b)$, yielding an explanation of HappelSeidel symmetry for a class of important Nakayama algebras. In particular, the weight sequence $(2,3,p)$ corresponds to an ADEchain, the $E_n$chain, extrapolating the exceptional Dynkin cases $E_6$, $E_7$ and $E_8$ to a whole sequence of triangulated categories.Advances in Mathematics. 03/2012; 237.  [Show abstract] [Hide abstract]
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 RingelSchmidmeier 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) CalabiYau categories, indexed by p, which naturally form an ADEchain.02/2010;  [Show abstract] [Hide abstract]
ABSTRACT: Let Λ be a tubular canonical algebra of quiver type. We describe an algorithm, which for numerical data computes all regular exceptional Λmodules, or more generally all indecomposable modules in exceptional tubes. The input for the algorithm is a quadruple consisting of the slope, the number of the tube, the quasisocle and the quasilength, the output are explicit matrices for the module with the data above.Journal of Algebra  J ALGEBRA. 01/2010; 323(10):27102734.  [Show abstract] [Hide abstract]
ABSTRACT: Let Λ be a tubular canonical algebra of quiver type. We describe an algorithm, which for numerical data computes all regular exceptional Λmodules, or more generally all indecomposable modules in exceptional tubes. The input for the algorithm is a quadruple consisting of the slope, the number of the tube, the quasisocle and the quasilength, the output are explicit matrices for the module with the data above.Journal of Algebra 01/2010; 323:27102734. · 0.58 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: We study the category C(X, Y) generated by an exceptional pair (X, Y) in a hereditary category H. If r = dimkHom(X, Y) ≥ 1 we show that there are exactly 3 possible types for C(X, Y), all derived equivalent to the category of finite dimensional modules mod(Hr) over the rKronecker algebra Hr. In general C(X, Y) will not be equivalent to a module category. More specifically, if H is the category of coherent sheaves over a weighted projective line , then C(X, Y) is equivalent to the category of coherent sheaves on the projective line 1 or to mod(Hr) and, if is wild, then every r ≥ 1 can occur in this way.Communications in Algebra 01/2009; 37(8):25472556. · 0.36 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: We describe explicitly all indecomposable modules of rank 6 over a domestic canonical algebra of quiver type over a field k of arbitrary characteristic. Together with the results given in [5] this yields an explicit description of all preprojective and preinjective indecomposable modules (and of all indecomposable modules if k is algebraically closed) for a domestic canonical algebra of quiver type. In particular for those algebras each preprojective and each preinjective indecomposable module can be represented by matrices whose coefficients are 0 and 1.International Journal of Algebra. 01/2008; 2:153161.  [Show abstract] [Hide abstract]
ABSTRACT: Let Λ be a tubular canonical algebra of quiver type over a field. We show that each exceptional Λmodule can be exhibited by matrices involving as coefficients 0, 1 and –1 if Λ is of type (3,3,3), (2,4,4) or (2,3,6) and by matrices involving as coefficients 0, 1, –1, λ, –λ and λ–1 if Λ is of type (2,2,2,2) and defined by a parameter λ.Algebras and Representation 09/2007; 10(5):481496. · 0.55 Impact Factor  [Show abstract] [Hide abstract]
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;  [Show abstract] [Hide abstract]
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. 01/2007;  [Show abstract] [Hide abstract]
ABSTRACT: BernsteinGelfandGelfand showed that the derived category of coherent sheaves on the projective nspace Db(cohPn) is equivalent to the stable ategory of the category of Zgraded finitedimensional modules over the exterior algebra. At the same time Beilinson gave a description of Db(cohPn) which can be interpreted in terms of tilting theory. We compare these two characterizations using Happel's description of the derived category of modules over a finitedimensional algebra of fnite global dimension.Communications in Algebra. 01/1992; 20(9):25132531.
Publication Stats
34  Citations  
2.60  Total Impact Points  
Top Journals
Institutions

2009–2014

University of Szczecin
 Institute of Mathematics
Stettin, West Pomeranian Voivodeship, Poland


2007

Universität Paderborn
 Department of Mathematics
Paderborn, North RhineWestphalia, Germany


1992

HumboldtUniversität zu Berlin
Berlín, Berlin, Germany
