- Søborg, Hovedstaden, Denmark
Ryszard Nest
Copenhagen University, Mathematics, Faculty Member
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We establish existence and stability results for solitons in noncommutative scalar field theories in even space dimension 2d. In particular, for any finite rank spectral projection P of the number operator N of the d-dimensional harmonic... more
We establish existence and stability results for solitons in noncommutative scalar field theories in even space dimension 2d. In particular, for any finite rank spectral projection P of the number operator N of the d-dimensional harmonic oscillator and sufficiently large noncommutativity parameter θ we prove the existence of a rotationally invariant soliton which depends smoothly on θ and converges to a multiple of P as θ →∞ . In the two-dimensional case we prove that these solitons are stable at large θ ,i f P = PN , where PN projects onto the space spanned by the N + 1 lowest eigenstates of N , and otherwise they are unstable. We also discuss the generalisation of the stability re- sults to higher dimensions. In particular, we prove stability of the soliton corresponding to P = P0 for all θ in its domain of existence. Finally, for arbitrary d and small values of θ , we prove without assuming rotational invariance that there do not exist any solitons depending smoothly on θ .
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We prove a $\unicode[STIX]{x1D6E4}$ -equivariant version of the algebraic index theorem, where $\unicode[STIX]{x1D6E4}$ is a discrete group of automorphisms of a formal deformation of a symplectic manifold. The particular cases of this... more
We prove a $\unicode[STIX]{x1D6E4}$ -equivariant version of the algebraic index theorem, where $\unicode[STIX]{x1D6E4}$ is a discrete group of automorphisms of a formal deformation of a symplectic manifold. The particular cases of this result are the algebraic version of the transversal index theorem related to the theorem of A. Connes and H. Moscovici for hypo-elliptic operators and the index theorem for the extension of the algebra of pseudodifferential operators by a group of diffeomorphisms of the underlying manifold due to A. Savin, B. Sternin, E. Schrohe and D. Perrot.
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Let $\Gamma$ be a finite dimensional Lie group and consider the smooth double loop group, i.e. the Fr\'echet Lie group of smooth maps from the 2-torus to $\Gamma$. For a finite dimensional Hilbert space V, let H denote the Hilbert... more
Let $\Gamma$ be a finite dimensional Lie group and consider the smooth double loop group, i.e. the Fr\'echet Lie group of smooth maps from the 2-torus to $\Gamma$. For a finite dimensional Hilbert space V, let H denote the Hilbert space of vector valued $L^2$-functions on the 2-torus. The purpose of this paper is to construct a higher central extension of the smooth double loop group from the representation of the smooth double loop group on H induced by a smooth action of $\Gamma$ on V. This higher central extension comes from an action of the smooth double loop group on a 2-category and yields a group cohomology class of degree 3 on the smooth double loop group. We show by a concrete computation that this group cohomology class is non-trivial in general. We relate our higher central extension to the Kac-Moody extension of the smooth single loop group as a higher dimensional analogue of the latter. More generally, given a group G acting on a bipolarised Hilbert space, we apply ...
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In this paper we introduce a notion of quantum Hamiltonian (co)action of Hopf algebras endowed with Drinfel'd twist structure (resp., 2-cocycles). First, we define a classical Hamiltonian action in the setting of Poisson Lie groups... more
In this paper we introduce a notion of quantum Hamiltonian (co)action of Hopf algebras endowed with Drinfel'd twist structure (resp., 2-cocycles). First, we define a classical Hamiltonian action in the setting of Poisson Lie groups compatible with the 2-cocycle stucture and we discuss a concrete example. This allows us to construct, out of the classical momentum map, a quantum momentum map in the setting of Hopf coactions and to quantize it by using Drinfel'd approach.
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We show that for a differential graded Lie algebra $\mathfrak{g}$ whose components vanish in degrees below -1 the nerve of the Deligne 2-groupoid is homotopy equivalent to the simplicial set of $\mathfrak{g}$-valued differential forms... more
We show that for a differential graded Lie algebra $\mathfrak{g}$ whose components vanish in degrees below -1 the nerve of the Deligne 2-groupoid is homotopy equivalent to the simplicial set of $\mathfrak{g}$-valued differential forms introduced by V.Hinich.
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We identify the 2-groupoid of deformations of a gerbe on a C manifold with the Deligne 2-groupoid of a corresponding twist of the DGLA of local Hochschild cochains on jets of C functions. Mathematics Subject Classification (2000). Primary... more
We identify the 2-groupoid of deformations of a gerbe on a C manifold with the Deligne 2-groupoid of a corresponding twist of the DGLA of local Hochschild cochains on jets of C functions. Mathematics Subject Classification (2000). Primary 53D55; Secondary 58J42, 18D05.
In this paper we compute the deformation theory of a special class of algebras, namely of Azumaya algebras on a manifold ($C^{\infty}$ or complex analytic).
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The main purpose of this paper is to modify the orbit method for the Baum-Connes conjecture as developed by Chabert, Echterhoff and Nest in their proof of the Connes-Kasparov conjecture for almost connected groups \cite{MR2010742} in... more
The main purpose of this paper is to modify the orbit method for the Baum-Connes conjecture as developed by Chabert, Echterhoff and Nest in their proof of the Connes-Kasparov conjecture for almost connected groups \cite{MR2010742} in order to deal with linear algebraic groups over local function fields (i.e., non-archimedean local fields of positive characteristic). As a consequence, we verify the Baum-Connes conjecture for certain Levi-decomposable linear algebraic groups over local function fields. One of these is the Jacobi group, which is the semidirect product of the symplectic group and the Heisenberg group.
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Let $\Gamma$ be a discrete icc subgroup of PSL(2,R) of infinite covolume. and let M denote the quotient of the unit disc by $\Gamma$. We prove that a Toeplitz operator with $\Gamma$-invariant symbol f in C(M) is Brauer Fredholm if its... more
Let $\Gamma$ be a discrete icc subgroup of PSL(2,R) of infinite covolume. and let M denote the quotient of the unit disc by $\Gamma$. We prove that a Toeplitz operator with $\Gamma$-invariant symbol f in C(M) is Brauer Fredholm if its symbol is invertible on the boundary of M and its Brauer index is equal to the winding number of f at the boundary. We construct the associated extension of the algebra of functions continuous on the boundary of M by the Brauer ideal in the C*-algebra generated by such operators.
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We extend the formality theorem of M. Kontsevich from deformations of the structure sheaf on a manifold to deformations of gerbes.
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We carefully define and study C � -algebras over topological spaces, possibly non- Hausdorff, and review some relevant results from point-set topology along the way. We explain the triangulated category structure on the bivariant Kasparov... more
We carefully define and study C � -algebras over topological spaces, possibly non- Hausdorff, and review some relevant results from point-set topology along the way. We explain the triangulated category structure on the bivariant Kasparov theory over a topo- logical space and study the analogue of the bootstrap class for C � -algebras over a finite topological space.
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We deduce the Riemann-Roch type formula expressing the microlocal Euler class of a perfect complex of D-modules in terms of the Chern character of the associated symbol complex and the Todd class of the manifold from the Riemann-Roch type... more
We deduce the Riemann-Roch type formula expressing the microlocal Euler class of a perfect complex of D-modules in terms of the Chern character of the associated symbol complex and the Todd class of the manifold from the Riemann-Roch type theorem for periodic cyclic cocycles of a symplectic deformation quantization. The proof of the latter is contained in the sequel to this paper.
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We prove a Riemann-Roch formula for deformation quantization of complex manifolds and its corollary, an index theorem for elliptic pairs conjectured by Schapira and Schneiders.
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Abstract. Let Г be a discrete subgroup of PSL (2, R) of infinite covolume with infinite conjugacy classes." Ht denotes the Hilbert space consisting of analytic functions in L2 (D,(Im z)'~ 2dzdz) and, for t> 1, nt denotes the... more
Abstract. Let Г be a discrete subgroup of PSL (2, R) of infinite covolume with infinite conjugacy classes." Ht denotes the Hilbert space consisting of analytic functions in L2 (D,(Im z)'~ 2dzdz) and, for t> 1, nt denotes the corresponding projective unitary representation of P5L (2, R) on this Hubert space. Let At be the J/oo factor given by the commutant of тг ((Г) in B (Ht). Let F denote a fundamental domain for Г in D. We assume that í> 5 and give дМ= Ж> Г¡ F the topology of disjoint union of its connected components. Suppose that/is a ...
We determine the additional structure which arises on the classical limit of a DQ-algebroid.
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In this paper we shall compute the cyclic cohomology of a non-commutative torus, i.e., a certain algebra associated with an antisymmetric bicharacter of a finite rank free abelian group G. The main result is 1.1 where The method of... more
In this paper we shall compute the cyclic cohomology of a non-commutative torus, i.e., a certain algebra associated with an antisymmetric bicharacter of a finite rank free abelian group G. The main result is 1.1 where The method of computation generalises the computation of the cyclic cohomology of the irrational rotation algebras given by Connes in [3]. (Our method works equally well also in the rational case, which was dealt with by a different method by Connes in [3].) We first describe the Hochschild cohomology of in an explicit way, and then combine this description with the exact sequence of [3]: 1.2
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... under colimits Arthur Bartels, Siegfried Echterhoff, and Wolfgang Liick 41 Coarse and equivariant co-assembly maps Heath Emerson and Ralf ... cyclic homology for quantum groups Christian Voigt 151 A Schwartz type algebra for the... more
... under colimits Arthur Bartels, Siegfried Echterhoff, and Wolfgang Liick 41 Coarse and equivariant co-assembly maps Heath Emerson and Ralf ... cyclic homology for quantum groups Christian Voigt 151 A Schwartz type algebra for the tangent groupoid Paulo Carrillo Rouse 181 C ...
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A simple proof of Bott periodicity, and at the same time of the Connes isomorphism theorem for one-parameter crossed products, is given using continuous fields ofC*-algebras.
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Let Gamma be a discrete subgroup of PSL (2; R) of infinite covolume withinfinite conjugacy classes. Let H t be the Hilbert space consisting of analyticfunctions in L2 (H;(Im z) tGamma2dzdz) and let t be the corresponding,... more
Let Gamma be a discrete subgroup of PSL (2; R) of infinite covolume withinfinite conjugacy classes. Let H t be the Hilbert space consisting of analyticfunctions in L2 (H;(Im z) tGamma2dzdz) and let t be the corresponding, projectiveunitary representation of PSL (2; R) on this Hilbert space, for t? 1. We denoteby A t the commutant of t (Gamma) in B (H t). Then A t is a II 1 factor. Assumethat F is a fundamental domain for Gamma in H. Let f be a Gamma-invariant function on H, that is continuous and invertible onthe closure of the fundamental ...
Let $\Gamma$ be a discrete icc subgroup of PSL(2,R) of infinite covolume. and let M denote the quotient of the unit disc by $\Gamma$. We prove that a Toeplitz operator with $\Gamma$-invariant symbol f in C(M) is Brauer Fredholm if its... more
Let $\Gamma$ be a discrete icc subgroup of PSL(2,R) of infinite covolume. and let M denote the quotient of the unit disc by $\Gamma$. We prove that a Toeplitz operator with $\Gamma$-invariant symbol f in C(M) is Brauer Fredholm if its symbol is invertible on the boundary of M and its Brauer index is equal to the winding number of f at the boundary. We construct the associated extension of the algebra of functions continuous on the boundary of M by the Brauer ideal in the C*-algebra generated by such operators.
We carefully define and study C*-algebras over topological spaces, possibly non-Hausdorff, and review some relevant results from point-set topology along the way. We explain the triangulated category structure on the bivariant Kasparov... more
We carefully define and study C*-algebras over topological spaces, possibly non-Hausdorff, and review some relevant results from point-set topology along the way. We explain the triangulated category structure on the bivariant Kasparov theory over a topological space. We introduce and describe an analogue of the bootstrap class for C*-algebras over a finite topological space.
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We investigate determinants of Koszul complexes of holomorphic functions of a commuting tuple of bounded operators acting on a Hilbert space. Our main result shows that the analytic joint torsion, which compares two such determinants, can... more
We investigate determinants of Koszul complexes of holomorphic functions of a commuting tuple of bounded operators acting on a Hilbert space. Our main result shows that the analytic joint torsion, which compares two such determinants, can be computed by a local formula which involves a tame symbol of the involved holomorphic functions. As an application we are able to extend the classical tame symbol of meromorphic functions on a Riemann surface to the more involved setting of transversal functions on a complex analytic curve. This follows by spelling out our main result in the case of Toeplitz operators acting on the Hardy space over the polydisc.
... Lately the interest in the non-commutative tori has been revived as they seem to appear naturally in string theory [CDS98]. ... zα from Aθ to C(TN ) extends to an isomorphism between the Hilbert spaces L2(Aθ,τ) and L2(TN ) when TN is... more
... Lately the interest in the non-commutative tori has been revived as they seem to appear naturally in string theory [CDS98]. ... zα from Aθ to C(TN ) extends to an isomorphism between the Hilbert spaces L2(Aθ,τ) and L2(TN ) when TN is equipped with the normalized Haar measure ...
We extend the formality theorem of Maxim Kontsevich from deformations of the structure sheaf on a manifold to deformations of gerbes on smooth and complex manifolds.
We extend the notion of Poincar\'e duality in KK-theory to the setting of quantum group actions. An important ingredient in our approach is the replacement of ordinary tensor products by braided tensor products. Along the way we discuss... more
We extend the notion of Poincar\'e duality in KK-theory to the setting of quantum group actions. An important ingredient in our approach is the replacement of ordinary tensor products by braided tensor products. Along the way we discuss general properties of equivariant KK-theory for locally compact quantum groups, including the construction of exterior products. As an example, we prove that the standard Podle\'s sphere is equivariantly Poincar\'e dual to itself.
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This paper establishes a link between noncommutative geometry and canonical quantum gravity. A semi-finite spectral triple over a space of connections is presented. The triple involves an algebra of holonomy loops and a Dirac-type... more
This paper establishes a link between noncommutative geometry and canonical quantum gravity. A semi-finite spectral triple over a space of connections is presented. The triple involves an algebra of holonomy loops and a Dirac-type operator, which resembles a global functional derivation operator. The commutation relation between the Dirac operator and the algebra has a structure related to the Poisson bracket of general relativity. Moreover, the associated Hilbert space corresponds, up to a certain symmetry group, to the Hilbert space of diffeomorphism-invariant states known from loop quantum gravity. Correspondingly, the square of the Dirac operator has, in terms of loop quantum gravity, the form of a global area-squared operator. Furthermore, the spectral action functional resembles a partition function of quantum gravity. The construction is background independent and is based on an inductive system of triangulations. This paper is the first of two papers on the subject.
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In this paper we consider deformations of an algebroid stack on an etale groupoid. We construct a differential graded Lie algebra (DGLA) which controls this deformation theory. In the case when the algebroid is a twisted form of functions... more
In this paper we consider deformations of an algebroid stack on an etale groupoid. We construct a differential graded Lie algebra (DGLA) which controls this deformation theory. In the case when the algebroid is a twisted form of functions we show that this DGLA is quasiisomorphic to the twist of the DGLA of Hochschild cochains on the algebra of functions
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We identify the 2-groupoid of deformations of a gerbe on a smooth manifold with the Deligne 2-groupoid of a corresponding twist of the DGLA of local Hochschild cochains on infinite jets of smooth functions.