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60
Noncommutative symplectic geometry, quiver varieties, and operads
"... to Liza Quiver varieties have recently appeared in various different areas of Mathematics such as representation theory of KacMoody algebras and quantum groups, instantons on 4manifolds, and resolutions Kleinian singularities. In this paper, we show that many important affine quiver varieties, e.g ..."
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Cited by 41 (7 self)
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to Liza Quiver varieties have recently appeared in various different areas of Mathematics such as representation theory of KacMoody algebras and quantum groups, instantons on 4manifolds, and resolutions Kleinian singularities. In this paper, we show that many important affine quiver varieties, e.g., the CalogeroMoser space, can be imbedded as coadjoint orbits in the dual of an appropriate infinite dimensional Lie algebra. In particular, there is an infinitesimally transitive action of the Lie algebra in question on the quiver variety. Our construction is based on an extension of Kontsevich’s formalism of ‘noncommutative Symplectic geometry’. We show that this formalism acquires its most adequate and natural formulation in the much more general framework of Pgeometry, a ‘noncommutative geometry ’ for an algebra over an arbitrary cyclic Koszul operad.
Twovector bundles and forms of elliptic cohomology
 in Topology, Geometry and Quantum Field Theory, LMS Lecture note series 308
, 2004
"... The work to be presented in this paper has been inspired by several of Professor Graeme Segal’s papers. Our search for a geometrically defined elliptic cohomology theory with associated elliptic objects obviously stems from his Bourbaki seminar [Se88]. Our readiness to form group completions of symm ..."
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Cited by 32 (4 self)
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The work to be presented in this paper has been inspired by several of Professor Graeme Segal’s papers. Our search for a geometrically defined elliptic cohomology theory with associated elliptic objects obviously stems from his Bourbaki seminar [Se88]. Our readiness to form group completions of symmetric monoidal categories
Pseudodifferential Operators on Manifolds with A LIE STRUCTURE AT INFINITY
, 2003
"... Several interesting examples of noncompact manifolds M0 whose geometry at infinity is described by Lie algebras of vector fields V ⊂ Γ(M; T M) (on a compactification of M0 to a manifold with corners M) were studied for instance in [28, 31, 46]. In [1], the geometry of manifolds described by Lie alg ..."
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Cited by 28 (13 self)
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Several interesting examples of noncompact manifolds M0 whose geometry at infinity is described by Lie algebras of vector fields V ⊂ Γ(M; T M) (on a compactification of M0 to a manifold with corners M) were studied for instance in [28, 31, 46]. In [1], the geometry of manifolds described by Lie algebras of vector fields – baptised “manifolds with a Lie structure at infinity ” there – was studied from an axiomatic point of view. In this paper, we define and study the algebra Ψ ∞ 1,0,V (M0), which is an algebra of pseudodifferential operators canonically associated to a manifold M0 with the Lie structure at infinity V ⊂ Γ(M; T M). We show that many of the properties of the usual algebra of pseudodifferential operators on a compact manifold extend to Ψ ∞ 1,0,V (M0). We also consider the algebra Diff ∗ V (M0) of differential operators on M0 generated by V and C ∞ (M), and show that Ψ ∞ 1,0,V (M0) is a “microlocalization” of Diff ∗ V (M0). We also define and study semiclassical and “suspended ” versions of the algebra Ψ ∞ 1,0,V (M0). Thus, our constructions solves a conjecture of Melrose [28].
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 22 (3 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.
Cyclic Cohomology of Étale Groupoids; The General Case
 Ktheory
, 1999
"... We give a general method for computing the cyclic cohomology of crossed products by 'etale groupoids, extending the FeiginTsyganNistor spectral sequences. In particular we extend the computations performed by Brylinski, Burghelea, Connes, Feigin, Karoubi, Nistor and Tsygan for the convolution alge ..."
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Cited by 21 (1 self)
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We give a general method for computing the cyclic cohomology of crossed products by 'etale groupoids, extending the FeiginTsyganNistor spectral sequences. In particular we extend the computations performed by Brylinski, Burghelea, Connes, Feigin, Karoubi, Nistor and Tsygan for the convolution algebra C 1 c (G) of an 'etale groupoid, removing the Hausdorffness condition and including the computation of hyperbolic components. Examples like group actions on manifolds and foliations are considered. Keywords: cyclic cohomology, groupoids, crossed products, duality, foliations. Contents 1 Introduction 3 2 Homology and Cohomology of Sheaves on ' Etale Groupoids 4 2.1 ' Etale Groupoids : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 2.2 \Gamma c in the nonHausdorff case : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 6 2.3 Homology and Cohomology of ' Etale Groupoids : : : : : : : : : : : : : : : : : : : : : 8 3 Cyclic Homologies of Sheaves ...
The periodic cyclic homology of IwahoriHecke algebras
, 2008
"... We determine the periodic cyclic homology of the IwahoriHecke algebras Hq, for q ∈ C∗ not a “proper root of unity.” (In this paper, by a proper root of unity we shall mean a root of unity other than 1.) Our method is based on a general result on periodic cyclic homology, which states that a “weakl ..."
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Cited by 17 (7 self)
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We determine the periodic cyclic homology of the IwahoriHecke algebras Hq, for q ∈ C∗ not a “proper root of unity.” (In this paper, by a proper root of unity we shall mean a root of unity other than 1.) Our method is based on a general result on periodic cyclic homology, which states that a “weakly spectrum preserving” morphism of finite type algebras induces an isomorphism in periodic cyclic homology. The concept of a weakly spectrum preserving morphism is defined in this paper, and most of our work is devoted to understanding this class of morphisms. Results of Kazhdan–Lusztig and Lusztig show that, for the indicated values of q, there exists a weakly spectrum preserving morphism φq: Hq → J, to a fixed finite type algebra J. This proves that φq induces an isomorphism in periodic cyclic homology and, in particular, that all algebras Hq have the same periodic cyclic homology, for the indicated values of q. The periodic cyclic homology groups of the algebra H1 can then be determined directly, using results of Karoubi and Burghelea, because it is the group algebra of an extended affine Weyl group.
Foliation groupoids and their cyclic homology
 Advances of Mathematics
"... The purpose of this paper is to prove two theorems which concern the position of étale groupoids among general smooth (or ”Lie”) groupoids. Our motivation comes from the noncommutative geometry and algebraic topology concerning leaf spaces of foliations. Here, one is concerned with invariants of th ..."
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Cited by 17 (6 self)
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The purpose of this paper is to prove two theorems which concern the position of étale groupoids among general smooth (or ”Lie”) groupoids. Our motivation comes from the noncommutative geometry and algebraic topology concerning leaf spaces of foliations. Here, one is concerned with invariants of the holonomy groupoid of a foliation
SPECTRAL SECTIONS AND HIGHER ATIYAHPATODISINGER INDEX THEORY ON GALOIS COVERINGS
 GAFA GEOMETRIC AND FUNCTIONAL ANALYSIS
, 1998
"... In this paper we consider Γ → ˜ M → M, a Galois covering with boundary and ˜ D/, a Γinvariant generalized Dirac operator on ˜M. We assume that the group Γ is of polynomial growth with respect to a word metric. By employing the notion of noncommutative spectral section associated to the boundary op ..."
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Cited by 13 (5 self)
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In this paper we consider Γ → ˜ M → M, a Galois covering with boundary and ˜ D/, a Γinvariant generalized Dirac operator on ˜M. We assume that the group Γ is of polynomial growth with respect to a word metric. By employing the notion of noncommutative spectral section associated to the boundary operator ˜ D/ 0 and the bcalculus on Galois coverings with boundary, we develop a higher AtiyahPatodiSinger index theory. Our main theorem extends to such ΓGalois coverings with boundary the higher index theorem of ConnesMoscovici.
Local index theory over etale groupoids
 J. Reine Angew. Math
"... Abstract. We give a superconnection proof of Connes ’ index theorem for proper cocompact actions of étale groupoids. This includes Connes ’ general foliation index theorem for foliations with Hausdorff holonomy groupoid. 1. ..."
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Cited by 13 (2 self)
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Abstract. We give a superconnection proof of Connes ’ index theorem for proper cocompact actions of étale groupoids. This includes Connes ’ general foliation index theorem for foliations with Hausdorff holonomy groupoid. 1.
The FrölicherNijenhuis Bracket In Non Commutative Differential Geometry
, 1993
"... this paper. We carry over to a quite general noncommutative setting some of the basic tools of differential geometry. From the very beginning we use the setting of convenient vector spaces developed by Frolicher and Kriegl. The reasons for this are the following: If the noncommutative theory shoul ..."
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Cited by 12 (7 self)
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this paper. We carry over to a quite general noncommutative setting some of the basic tools of differential geometry. From the very beginning we use the setting of convenient vector spaces developed by Frolicher and Kriegl. The reasons for this are the following: If the noncommutative theory should contain some version of differential geometry, a manifold M should be represented by the algebra C