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133
Formal (non)commutative symplectic geometry, from
 The Gelfand Mathematical Seminars, 1990–1992”, Birkhäuser
, 1993
"... and I had tried to understand a remark of J. Stasheff [15] on open string theory and higher associative algebras [16]. Then I found a strange construction of cohomology classes of mapping class groups using as initial data any differential graded algebra with finitedimensional cohomology and a kin ..."
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Cited by 150 (4 self)
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and I had tried to understand a remark of J. Stasheff [15] on open string theory and higher associative algebras [16]. Then I found a strange construction of cohomology classes of mapping class groups using as initial data any differential graded algebra with finitedimensional cohomology and a kind of Poincare ́ duality. Later generalizations to the commutative and Lie cases appeared. In attempts to formulate all this I have developed a kind of (non)commutative calculus. The commutative version has fruitful applications in topology of smooth manifolds in dimensions ≥ 3. The beginnings of applications are perturbative ChernSimons theory (S. Axelrod and I.M. Singer [1] and myself), V. Vassiliev’s theory of knot invariants and discriminants (see [19], new results in [2]) and V. Drinfeld’s works on quasiHopf algebras (see [6]), also containing elements of Lie calculus. Here I present the formal aspects of the story. Theorem 1.1 is the main motivation for my interest in noncommutative symplectic geometry. Towards the end the exposition becomes a bit more vague and informal. Nevertheless, I hope that I will convince the reader that noncommutative calculus has every right to exist. I have benefited very much from conversations with B. Feigin, V. Retakh,
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 62 (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 37 (18 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.
Noncommutative geometry of finite groups
 J. Phys. A
, 1996
"... A finite set can be supplied with a group structure which can then be used to select (classes of) differential calculi on it via the notions of left, right and bicovariance. A corresponding framework has been developed by Woronowicz, more generally for Hopf algebras including quantum groups. A dif ..."
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Cited by 30 (3 self)
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A finite set can be supplied with a group structure which can then be used to select (classes of) differential calculi on it via the notions of left, right and bicovariance. A corresponding framework has been developed by Woronowicz, more generally for Hopf algebras including quantum groups. A differential calculus is regarded as the most basic structure needed for the introduction of further geometric notions like linear connections and, moreover, for the formulation of field theories and dynamics on finite sets. Associated with each bicovariant first order differential calculus on a finite group is a braid operator which plays an important role for the construction of distinguished geometric structures. For a covariant calculus, there are notions of invariance for linear connections and tensors. All these concepts are explored for finite groups and illustrated with examples. Some results are formulated more generally for arbitrary associative (Hopf) algebras. In particular, the problem of extension of a connection on a bimodule (over an associative algebra) to tensor products is investigated, leading to the class of ‘extensible connections’. It is shown that invariance properties of an extensible connection on a bimodule over a Hopf algebra are carried over to the extension. Furthermore, an invariance property of a connection is also shared by a ‘dual connection ’ which exists on the dual bimodule (as defined in this work).2 1
Universal qDifferential Calculus and qAnalog of Homological Algebra
 Acta Math. Univ. Comenian
, 1996
"... . We recall the definition of qdifferential algebras and discuss some representative examples. In particular we construct the qanalog of the Hochschild coboundary. We then construct the universal qdifferential envelope of a unital associative algebra and study its properties. The paper also conta ..."
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Cited by 29 (18 self)
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. We recall the definition of qdifferential algebras and discuss some representative examples. In particular we construct the qanalog of the Hochschild coboundary. We then construct the universal qdifferential envelope of a unital associative algebra and study its properties. The paper also contains general results on d N = 0. 1. Introduction and Algebraic Preliminaries At the origin of this paper there is the longstanding physicallymotivated interest of one of the authors (R.K.) on Z 3 graded structures and differential calculi [RK] although here the point of view is somehow different. There is also the observation that the simplicial (co)homology admits Z N versions leading to cyclotomic homology [Sark] and that, more generally, this suggests that one can introduce "qanalog of homological algebra" for each primitive root q of the unity [Kapr]. Moreover the occurrence of various notions of "qanalog" in connection with quantum groups suggests to include in the formulation t...
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 26 (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