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163
Formal (non)commutative symplectic geometry
 THE GELFAND MATHEMATICAL SEMINARS, 1990–1992”, BIRKHÄUSER
, 1993
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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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Cited by 156 (1 self)
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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].
Projective module description of the qmonopole
 COMUN.MATH.PHYS
, 1999
"... The Dirac qmonopole connection is used to compute projector matrices of quantum Hopf line bundles for arbitrary winding number. The ChernConnes pairing of cyclic cohomology and Ktheory is computed for the winding number −1. The nontriviality of this pairing is used to conclude that the quantum p ..."
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Cited by 50 (20 self)
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The Dirac qmonopole connection is used to compute projector matrices of quantum Hopf line bundles for arbitrary winding number. The ChernConnes pairing of cyclic cohomology and Ktheory is computed for the winding number −1. The nontriviality of this pairing is used to conclude that the quantum principal Hopf fibration is noncleft. Among general results, we provide a leftright symmetric characterization of the canonical strong connections on quantum principal homogeneous spaces with an injective antipode. We also provide for arbitrary strong connections on algebraic quantum principal bundles (HopfGalois extensions) their associated covariant derivatives on projective modules.
Lectures on graded differential algebras and noncommutative geometry
, 1999
"... These notes contain a survey of some aspects of the theory of graded differential algebras and of noncommutative differential calculi as well as of some applications connected with physics. They also give a description of several new developments. ..."
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Cited by 49 (5 self)
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These notes contain a survey of some aspects of the theory of graded differential algebras and of noncommutative differential calculi as well as of some applications connected with physics. They also give a description of several new developments.
Notes on A∞algebras, A∞categories and noncommutative geometry, Homological mirror symmetry
 Lecture Notes in Phys
, 2009
"... 1.1 A∞algebras as spaces........................ 2 ..."
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Cited by 43 (0 self)
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1.1 A∞algebras as spaces........................ 2
Noncommutative geometry based on commutator expansions
 J. Reine Angew. Math
, 1998
"... Contents 3. The NCaffine space and FeynmanMaslov operator calculus. 4. Detailed study of algebraic NCmanifolds. 5. Examples of NCmanifolds. The term “noncommutative geometry ” has come to signify a vast framework of ideas ..."
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Cited by 36 (0 self)
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Contents 3. The NCaffine space and FeynmanMaslov operator calculus. 4. Detailed study of algebraic NCmanifolds. 5. Examples of NCmanifolds. The term “noncommutative geometry ” has come to signify a vast framework of ideas