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172
CalabiYau algebras
"... Abstract. We introduce some new algebraic structures arising naturally in the geometry of CY manifolds and mirror symmetry. We give a universal construction of CY algebras in terms of a noncommutative symplectic DG algebra resolution. In dimension 3, the resolution is determined by a noncommutative ..."
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Abstract. We introduce some new algebraic structures arising naturally in the geometry of CY manifolds and mirror symmetry. We give a universal construction of CY algebras in terms of a noncommutative symplectic DG algebra resolution. In dimension 3, the resolution is determined by a noncommutative potential. Representation varieties of the CY algebra are intimately related to the set of critical points, and to the sheaf of vanishing cycles of the potential. Numerical invariants, like ranks of cyclic homology groups, are expected to be given by ‘matrix integrals ’ over representation varieties. We discuss examples of CY algebras involving quivers, 3dimensional McKay correspondence, crepant resolutions, Sklyanin algebras, hyperbolic 3manifolds and ChernSimons. Examples related to quantum Del Pezzo surfaces are discussed in [EtGi].
Noncommutative FiniteDimensional Manifolds  I. SPHERICAL MANIFOLDS AND RELATED EXAMPLES
, 2001
"... We exhibit large classes of examples of noncommutative finitedimensional manifolds which are (nonformal) deformations of classical manifolds. The main result of this paper is a complete description of noncommutative threedimensional spherical manifolds, a noncommutative version of the sphere S 3 d ..."
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Cited by 125 (15 self)
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We exhibit large classes of examples of noncommutative finitedimensional manifolds which are (nonformal) deformations of classical manifolds. The main result of this paper is a complete description of noncommutative threedimensional spherical manifolds, a noncommutative version of the sphere S 3 defined by basic Ktheoretic equations. We find a 3parameter family of deformations of the standard 3sphere S 3 and a corresponding 3parameter deformation of the 4dimensional Euclidean space R 4. For generic values of the deformation parameters we show that the obtained algebras of polynomials on the deformed R 4 u are isomorphic to the algebras introduced by Sklyanin in connection with the YangBaxter equation. Special values of the deformation parameters do not give rise to Sklyanin algebras and we extract a subclass, the θdeformations, which we generalize in any dimension and various contexts, and study in some details. Here, and
Mirror symmetry for weighted projective planes and their noncommutative deformations
, 2004
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Noncommutative curves and noncommutative surfaces
 Bulletin of the American Mathematical Society
"... Abstract. In this survey article we describe some geometric results in the theory of noncommutative rings and, more generally, in the theory of abelian categories. Roughly speaking and by analogy with the commutative situation, the category of graded modules modulo torsion over a noncommutative grad ..."
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Cited by 91 (7 self)
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Abstract. In this survey article we describe some geometric results in the theory of noncommutative rings and, more generally, in the theory of abelian categories. Roughly speaking and by analogy with the commutative situation, the category of graded modules modulo torsion over a noncommutative graded ring of quadratic, respectively cubic growth should be thought of as the noncommutative analogue of a projective curve, respectively surface. This intuition has lead to a remarkable number of nontrivial insights and results in noncommutative algebra. Indeed, the problem of classifying noncommutative curves (and noncommutative graded rings of quadratic growth) can be regarded as settled. Despite the fact that no classification of noncommutative surfaces is in sight, a rich body of nontrivial examples and techniques, including blowing
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 82 (21 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.
Graded Calabi Yau algebras of dimension 3
, 2006
"... In this paper we prove that Graded Calabi Yau Algebras of dimension 3 are isomorphic to path algebras of quivers with relations derived from a superpotential. We show that for a given quiver Q and a degree d, the set of good superpotentials of degree d, i.e. those that give rise to Calabi Yau alge ..."
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Cited by 70 (6 self)
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In this paper we prove that Graded Calabi Yau Algebras of dimension 3 are isomorphic to path algebras of quivers with relations derived from a superpotential. We show that for a given quiver Q and a degree d, the set of good superpotentials of degree d, i.e. those that give rise to Calabi Yau algebras is either empty or almost everything (in the measure theoretic sense). We also give some constraints on the structure of quivers that allow good superpotentials, and for the simplest quivers we give a complete list of the degrees for which good superpotentials exist.
A class of algebras similar to the enveloping algebra of sl2
 Transactions of the American Mathematical Society
, 1980
"... JSTOR is a notforprofit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JS ..."
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Cited by 55 (1 self)
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JSTOR is a notforprofit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. American Mathematical Society is collaborating with JSTOR to digitize, preserve and extend access to Transactions of the American Mathematical Society.
Noncommutative instantons and twistor transform
 Commun. Math. Phys
"... Recently N. Nekrasov and A. Schwarz proposed a modification of the ADHM construction of instantons which produces instantons on a noncommutative deformation of R4. In this paper we study the relation between their construction and algebraic bundles on noncommutative projective spaces. We exhibit one ..."
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Cited by 54 (4 self)
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Recently N. Nekrasov and A. Schwarz proposed a modification of the ADHM construction of instantons which produces instantons on a noncommutative deformation of R4. In this paper we study the relation between their construction and algebraic bundles on noncommutative projective spaces. We exhibit onetoone correspondences between three classes of objects: framed bundles on a noncommutative P2, certain complexes of sheaves on a noncommutative P3, and the modified ADHM data. The modified ADHM construction itself is interpreted in terms of a noncommutative version of the twistor transform. We also prove that the moduli space of framed bundles on the noncommutative P2 has a natural hyperkähler metric and is isomorphic as a hyperkähler manifold to the moduli space of framed torsion free sheaves on the commutative P2. The natural complex structures on the two moduli spaces do not coincide but are related by an SO(3) rotation. Finally, we propose a construction of instantons on a more general noncommutative R4 than the one considered by Nekrasov and Schwarz (a q – deformed R4).
Koszul and Gorenstein Properties for Homogeneous Algebras
, 2003
"... Koszul property was generalized to homogeneous algebras of degree N> 2 in [5], and related to Ncomplexes in [7]. We show that if the Nhomogeneous algebra A is generalized Koszul, ASGorenstein and of finite global dimension, then one can apply the Van den Bergh duality theorem [23] to A, i.e., ..."
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Cited by 44 (2 self)
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Koszul property was generalized to homogeneous algebras of degree N> 2 in [5], and related to Ncomplexes in [7]. We show that if the Nhomogeneous algebra A is generalized Koszul, ASGorenstein and of finite global dimension, then one can apply the Van den Bergh duality theorem [23] to A, i.e., there is a Poincaré duality between Hochschild homology and cohomology of A, as for N = 2.