Publications (11)5.21 Total impact
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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 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. · 1.37 Impact Factor  [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.Journal für die reine und angewandte Mathematik (Crelles Journal) 02/2010; · 1.08 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.Colloquium Mathematicum 01/2007; · 0.40 Impact Factor  [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; · 0.53 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: We investigate complete exceptional sequences E=(E 1,,E n ) in the derived category D b of finitedimensional 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 multitranslation (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 representationfinite.Algebras and Representation 01/2002; 5(2):201209. · 0.55 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: We show that up to a translation each automorphism of the derived category DX of coherent sheaves on a weighted projective line X, equivalently of the derived category DA of finite dimensional modules over a derived canonical algebra A, is composed of tubular mutations and automorphisms of X. In the case of genus one this implies that the automorphism group is a semidirect product of the braid group on three strands by a finite group. Moreover we prove that most automorphisms lift from the Grothendieck group to the derived category.Communications in Algebra 01/2000; 28(4):16851700. · 0.36 Impact Factor 
Article: Sheaves on a weighted projective line of genus one and representations of a tubular algebra
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ABSTRACT: The paper is devoted to the classification of indecomposable vector bundles over a weighted projective line 𝕏 of genus one. The authors focus on the semistability of indecomposable bundles on 𝕏 considered by W. Geigle and H. Lenzing [in Singularities, representations of algebras and vector bundles, Proc. Symp., Lambrecht 1985, Lect. Notes Math. 1273, 265297 (1987; Zbl 0651.14006)]. They use a Ktheoretic version of the RiemannRoch theorem and develop the theory of mutations. The paper focusses on mutations with respect to a non exceptional object. The role of shrinking functors [see C. M. Ringel, “Tame algebras and integral quadratic forms”, Lect. Notes Math. 1099 (1984; Zbl 0546.16013)] is taken up by mutations and line bundle shifts. There are described classifications of indecomposable sheaves and of finite dimensional right Λmodules, where Λ is viewed as an algebra with several objects. Among various applications we mention the introduction and the study of oneparameter families of sheaves and modules. We mention the existence of a class of indecomposable generic quasicoherent sheaves G q , satisfying specific conditions, on weighted projectives lines of genus one. If G q is defined on 𝕏 then End 𝕏 (G q )≅k(X) and Ext 𝕏 1 (G q ,G q )=0 for each q∈ℚ∪{∞}. The same methods can be used for the category Qcoh(𝕋), where 𝕋 is an elliptic curve and to the category Mod(Λ), where Λ is a canonical algebra of tubular type. The paper contains many historical remarks with explicit references.  [Show abstract] [Hide abstract]
ABSTRACT: The objective of this paper is to study a new class of algebras that generalise the classes of tame concealed algebras, tubular algebras and canonical algebras, all of which have been extensively studied in the representation theory of Artin algebras. Let thus k be an algebraically closed field and 𝕏=𝕏(p,λ ̲) be the weighted projective line associated to a weight sequence p=(p 1 ,⋯,p t ) of integers p i ≥1, and a parameter sequence λ ̲=(λ 1 ,⋯,λ t ) of pairwise distinct elements of ℙ 1 (k). A finite dimensional algebra Σ is called concealedcanonical (or almost concealed canonical) if Σ≅End 𝕏 T, where T is a tilting sheaf (or tilting bundle, respectively). Such an algebra is derivedequivalent to the associated canonical algebra Λ=Λ(p,λ ̲), it is also a quasitilted algebra. The authors prove that the representation type of Σ is entirely determined by its quadratic form q Σ , or, equivalently, by the invariant δ Σ =(t2)∑ i=1 t 1 p i , induced from the weight type (p,λ ̲) of Σ (which is shown to be uniquely determined, up to equivalence, by Σ). They also study the relation between the concepts of concealedcanonical and almost concealedcanonical, showing for instance that an algebra Σ is concealedcanonical if and only if both Σ and Σ op are almost concealedcanonical. If Σ has tubular weight type, these two notions coincide, and then Σ is a tubular algebra. More generally, an algebra is almost concealedcanonical if and only if it is a branch coenlargement of a concealedcanonical algebra. The authors also study the structure of the module category of an almost concealedcanonical algebra, showing that it splits naturally into four parts, which reduce to three if the algebra is actually concealedcanonical. They also determine, in this latter case, the existence of separating tubular families.
Publication Stats
92  Citations  
5.21  Total Impact Points  
Top Journals
Institutions

2009

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


2002–2007

Universität Paderborn
 Department of Mathematics
Paderborn, North RhineWestphalia, Germany


2000

Technische Universität Chemnitz
 Department of Mathematics
Chemnitz, Saxony, Germany
