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97
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 62 (9 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.
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 53 (20 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 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.
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 49 (8 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
R/Z index theory
 Comm. Anal. Geom
, 1994
"... We define topological and analytic indices in R/Z Ktheory and show that they are equal. 1 ..."
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Cited by 46 (4 self)
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We define topological and analytic indices in R/Z Ktheory and show that they are equal. 1
DEFORMATION THEORY OF REPRESENTATIONS OF PROP(ERAD)S I
"... Abstract. In this paper and its followup [MV08], we study the deformation theory of morphisms of properads and props thereby extending Quillen’s deformation theory for commutative rings to a nonlinear framework. The associated chain complex is endowed with an L∞algebra structure. Its MaurerCarta ..."
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Cited by 32 (7 self)
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Abstract. In this paper and its followup [MV08], we study the deformation theory of morphisms of properads and props thereby extending Quillen’s deformation theory for commutative rings to a nonlinear framework. The associated chain complex is endowed with an L∞algebra structure. Its MaurerCartan elements correspond to deformed structures, which allows us to give a geometric interpretation of these results.
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 ..."
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Cited by 27 (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 ...
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 27 (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
Bordism, rhoinvariants and the Baum–Connes conjecture
 J. NONCOMMUT. GEOM.
, 2007
"... Let � be a finitely generated discrete group. In this paper we establish vanishing results for rhoinvariants associated to (i) the spin Dirac operator of a spin manifold with positive scalar curvature and fundamental group �; (ii) the signature operator of the disjoint union of a pair of homotopy e ..."
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Cited by 23 (8 self)
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Let � be a finitely generated discrete group. In this paper we establish vanishing results for rhoinvariants associated to (i) the spin Dirac operator of a spin manifold with positive scalar curvature and fundamental group �; (ii) the signature operator of the disjoint union of a pair of homotopy equivalent oriented manifolds with fundamental group �. The invariants we consider are more precisely theAtiyah–Patodi–Singer ( APS) rhoinvariant associated to a pair of finite dimensional unitary representations 1; 2 W � ! U.d/, the L 2rhoinvariant of Cheeger–Gromov, the delocalized etainvariant of Lott for a nontrivial conjugacy class of � which is finite. We prove that all these rhoinvariants vanish if the group � is torsionfree and the Baum–Connes map for the maximal group C*algebra is bijective. This condition is satisfied, for example, by torsionfree amenable groups or by torsionfree discrete subgroups of SO.n; 1 / and SU.n; 1/. For the delocalized invariant we only assume the validity of the Baum–Connes conjecture for the reduced C*algebra. In addition to the examples above, this condition is satisfied e.g. by Gromov hyperbolic groups or by cocompact discrete subgroups of SL.3; C/. In particular, the three rhoinvariants associated to the signature operator are, for such groups, homotopy invariant. For the APS and the Cheeger–Gromov rhoinvariants the latter result had been established by Navin Keswani. Our proof reestablishes this result and also extends it to the delocalized etainvariant of Lott. The proof exploits in a fundamental way results from bordism theory as well as various generalizations of the APSindex theorem; it also embeds these results in general vanishing phenomena for degree zero higher rhoinvariants (taking values in A=ŒA; A � for suitable C*algebras A). We also obtain precise information about the etainvariants in question themselves, which are usually much more subtle objects than the rhoinvariants.
Dirac index classes and the noncommutative spectral flow
, 2003
"... We present a detailed proof of the existencetheorem for noncommutative spectral sections (see the noncommutative spectral flow, unpublished preprint, 1997). We apply this result to various indextheoretic situations, extending to the noncommutative context results of Booss– Wojciechowski, Melrose–P ..."
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Cited by 23 (6 self)
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We present a detailed proof of the existencetheorem for noncommutative spectral sections (see the noncommutative spectral flow, unpublished preprint, 1997). We apply this result to various indextheoretic situations, extending to the noncommutative context results of Booss– Wojciechowski, Melrose–Piazza and Dai–Zhang. In particular, we prove a variational formula, in K ðC n r ðGÞÞ; for the index classes associated to 1parameter family of Dirac operators on a Gcovering with boundary; this formula involves a noncommutative spectral flow for the boundary family. Next, we establish an additivity result, in K ðC * n r ðGÞÞ; for the index class defined by a Diractype operator associated to a closed manifold M and a map r: MBG when we assume that M is the union along a hypersurface F of two manifolds with boundary M Mþ,F M: Finally, we prove a defect formula for the signatureindex classes of two cutandpaste equivalent pairs ðM1; r1: M1BGÞ and ðM2; r2: M2BGÞ; where M1 Mþ, ðF;f1Þ M; M2 Mþ, ðF;f2Þ M and f jADiffðFÞ: The formula involves the noncommutative spectral flow of a suitable 1parameter family of twisted signature operators on F: We give applications to the problem of cutandpaste invariance of Novikov’s higher signatures on closed oriented manifolds.