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72
Derived categories of coherent sheaves and triangulated categories of singularities
, 2005
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Higher dimensional AuslanderReiten theory on maximal orthogonal subcategories
, 2005
"... We introduce the concept of maximal orthogonal subcategories over artin algebras and orders, and develop higher AuslanderReiten theory on them. ..."
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Cited by 42 (13 self)
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We introduce the concept of maximal orthogonal subcategories over artin algebras and orders, and develop higher AuslanderReiten theory on them.
Semicanonical bases and preprojective algebras
 Ann. Sci. École Norm. Sup
"... Abstract. Let n be a maximal nilpotent subalgebra of a complex simple Lie algebra of type A, D,E. Lusztig has introduced a basis of U(n) called the semicanonical basis, whose elements can be seen as certain constructible functions on varieties of modules over a preprojective algebra of the same Dynk ..."
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Cited by 38 (8 self)
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Abstract. Let n be a maximal nilpotent subalgebra of a complex simple Lie algebra of type A, D,E. Lusztig has introduced a basis of U(n) called the semicanonical basis, whose elements can be seen as certain constructible functions on varieties of modules over a preprojective algebra of the same Dynkin type as n. We prove a formula for the product of two elements of the dual of this semicanonical basis, and more generally for the product of two evaluation forms associated to arbitrary modules over the preprojective algebra. This formula plays an important role in our work on the relationship between semicanonical bases, representation theory of preprojective algebras, and Fomin and Zelevinsky’s theory of cluster algebras. It was inspired by recent results of Caldero and Keller. 1. Introduction and
Ideal classes of the Weyl algebra and noncommutative projective geometry
, 2001
"... Let R be the set of isomorphism classes of ideals in the Weyl algebra A = A1(C), and let C be the set of isomorphism classes of triples (V, X, Y), where V is a finitedimensional (complex) vector space, and X, Y are endomorphisms of V such that [X, Y]+I has rank 1. Following a suggestion of L. Le B ..."
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Cited by 22 (2 self)
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Let R be the set of isomorphism classes of ideals in the Weyl algebra A = A1(C), and let C be the set of isomorphism classes of triples (V, X, Y), where V is a finitedimensional (complex) vector space, and X, Y are endomorphisms of V such that [X, Y]+I has rank 1. Following a suggestion of L. Le Bruyn, we define a map θ: R → C by appropriately extending an ideal of A to a sheaf over a quantum projective plane, and then using standard methods of homological algebra. We prove that θ is inverse to a bijection ω: C → R constructed in [BW] by a completely different method. The main step in the proof is to show that θ is equivariant with respect to natural actions of the group G = Aut(A) on R and C: for that we have to study also the extensions of an ideal to certain weighted quantum projective planes. Along the way, we find an elementary description of θ.
Tame and wild projective curves and classification of vector bundles
 54. MR 1872612
"... Abstract. We propose a new method of classifying vector bundles on projective curves, especially singular ones, according to their “representation type”. In particular, we prove that the classification problem of vector bundles, respectively of torsion–free sheaves on projective curves is always e ..."
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Cited by 19 (5 self)
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Abstract. We propose a new method of classifying vector bundles on projective curves, especially singular ones, according to their “representation type”. In particular, we prove that the classification problem of vector bundles, respectively of torsion–free sheaves on projective curves is always either finite, or tame, or wild. We completely classify curves which are of finite, respectively tame, vector bundle type by their dual graph. Moreover, our methods yield a geometric description of all indecomposable vector bundles and torsion–free sheaves on finite and tame curves.
Indecomposable parabolic bundles and the existence of matrices in prescribed conjugacy class closures with product equal to the identity
, 2003
"... We study the possible dimension vectors of indecomposable parabolic bundles on the projective line, and use our answer to solve the problem of characterizing those collections of conjugacy classes of n × n matrices for which one can find matrices in their closures whose product is equal to the ide ..."
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Cited by 16 (4 self)
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We study the possible dimension vectors of indecomposable parabolic bundles on the projective line, and use our answer to solve the problem of characterizing those collections of conjugacy classes of n × n matrices for which one can find matrices in their closures whose product is equal to the identity matrix. Both answers depend on the root system of a KacMoody Lie algebra. Our proofs use Ringel’s theory of tubular algebras, work of Mihai on the existence of logarithmic connections, the RiemannHilbert correspondence and an algebraic version, due to Dettweiler and Reiter, of Katz’s middle convolution operation.
Noncommutative projective curves and quantum loop algebras
 Duke Math. J
, 2004
"... Abstract. We show that the Hall algebra of the category of coherent sheaves on a weighted projective line over a finite field provides a realization of the (quantized) enveloping algebra of a certain nilpotent subalgebra of the affinization of the corresponding KacMoody algebra. In particular, this ..."
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Cited by 16 (5 self)
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Abstract. We show that the Hall algebra of the category of coherent sheaves on a weighted projective line over a finite field provides a realization of the (quantized) enveloping algebra of a certain nilpotent subalgebra of the affinization of the corresponding KacMoody algebra. In particular, this yields a geometric realization of the quantized enveloping algebra of elliptic (or 2toroidal) algebras of types D (1,1)
Lectures on Hall algebras
, 2006
"... These notes represent the written, expanded and improved version of a series of lectures given at the winter school “Representation theory and related topics ” held at the ICTP in Trieste in January 2006. The topic for the lectures was “Hall algebras” and I have tried to give a survey of what I beli ..."
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Cited by 12 (0 self)
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These notes represent the written, expanded and improved version of a series of lectures given at the winter school “Representation theory and related topics ” held at the ICTP in Trieste in January 2006. The topic for the lectures was “Hall algebras” and I have tried to give a survey of what I believe are the most fundamental