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European Mathematical Society Publishing House
2024-03-29 02:13:41
44
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=JNCG&vol=11&update_since=2024-03-29
Journal of Noncommutative Geometry
J. Noncommut. Geom.
JNCG
1661-6952
1661-6960
Global analysis, analysis on manifolds
General
10.4171/JNCG
http://www.ems-ph.org/doi/10.4171/JNCG
subscribers, moving wall 5 years
European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
11
2017
1
Bost–Connes systems, categorification, quantum statistical mechanics, and Weil numbers
Matilde
Marcolli
California Institute of Technology, PASADENA, UNITED STATES
Gonçalo
Tabuada
Massachusetts Institute of Technology, CAMBRIDGE, UNITED STATES
Quantum statistical mechanical systems, Gibbs states, zeta function, polylogarithms, Tannakian categories, Weil numbers, motives, Weil restriction
In this article we develop a broad generalization of the classical Bost–Connes system, where roots of unity are replaced by an algebraic datum consisting of an abelian group and a semi-group of endomorphisms. Examples include roots of unity, Weil restriction, algebraic numbers,Weil numbers, CM fields, germs, completion ofWeil numbers, etc. Making use of the Tannakian formalism, we categorify these algebraic data. For example, the categorification of roots of unity is given by a limit of orbit categories of Tate motives while the categorification of Weil numbers is given by Grothendieck’s category of numerical motives over a finite field. To some of these algebraic data (e.g. roots of unity, algebraic numbers, Weil numbers, etc), we associate also a quantum statistical mechanical system with several remarkable properties, which generalize those of the classical Bost–Connes system. The associated partition function, low temperature Gibbs states, and Galois action on zero-temperature states are then studied in detail. For example, we show that in the particular case of the Weil numbers the partition function and the low temperature Gibbs states can be described as series of polylogarithms.
Number theory
Algebraic geometry
Statistical mechanics, structure of matter
1
49
10.4171/JNCG/11-1-1
http://www.ems-ph.org/doi/10.4171/JNCG/11-1-1
Some quasitensor autoequivalences of Drinfeld doubles of finite groups
Peter
Schauenburg
Université de Bourgogne, DIJON, FRANCE
Drinfeld double, fusion categories, modular categories, braided tensor categories
We report on two classes of autoequivalences of the category of Yetter–Drinfeld modules over a finite group, or, equivalently the Drinfeld center of the category of representations of a finite group. Both operations are related to the $r$-th power operation, with $r$ relatively prime to the exponent of the group. One is defined more generally for the group-theoretical fusion category defined by a finite group and an arbitrary subgroup, while the other seems particular to the case of Yetter–Drinfeld modules. Both autoequivalences preserve higher Frobenius–Schur indicators up to Galois conjugation, and they preserve tensor products, although neither of them can in general be endowed with the structure of a monoidal functor.
Category theory; homological algebra
Associative rings and algebras
Group theory and generalizations
51
70
10.4171/JNCG/11-1-2
http://www.ems-ph.org/doi/10.4171/JNCG/11-1-2
Hopf algebras and universal Chern classes
Henri
Moscovici
Ohio State University, COLUMBUS, UNITED STATES
Bahram
Rangipour
University of New Brunswick, FREDERICTON, NB, CANADA
Hopf cyclic cohomology, universal Chern classes, characteristic classes of foliations, Connes–Moscovici Hopf algebras
We construct a variant $\mathcal K_n$ of the Hopf algebra $\mathcal H_n$, which acts directly on the noncommutative model for the space of leaves of a general foliation rather than on its frame bundle. We prove that the Hopf cyclic cohomology of $\mathcal K_n$ is isomorphic to that of the pair $(\mathcal H_n, \mathfrak {gl}_n)$ and thus consists of the universal Hopf cyclic Chern classes. We also realize these classes in terms of geometric cocycles.
Associative rings and algebras
Number theory
Global analysis, analysis on manifolds
71
109
10.4171/JNCG/11-1-3
http://www.ems-ph.org/doi/10.4171/JNCG/11-1-3
The derived non-commutative Poisson bracket on Koszul Calabi–Yau algebras
Xiaojun
Chen
Sichuan University, CHENGDU, CHINA
Alimjon
Eshmatov
University of Western Ontario, LONDON, CANADA
Farkhod
Eshmatov
Sichuan University, CHENGDU, CHINA
Song
Yang
Sichuan University, CHENGDU, CHINA
Noncommutative Poisson structure, Calabi–Yau algebra, cyclic homology
Let $A$ be a Koszul (or more generally, $N$-Koszul) Calabi–Yau algebra. Inspired by the works of Kontsevich, Ginzburg and Van den Bergh, we show that there is a derived non-commutative Poisson structure on $A$, which induces a graded Lie algebra structure on the cyclic homology of $A$; moreover, we show that the Hochschild homology of $A$ is a Lie module over the cyclic homology and the Connes long exact sequence is in fact a sequence of Lie modules. Finally, we show that the Leibniz–Loday bracket associated to the derived non-commutative Poisson structure on $A$ is naturally mapped to the Gerstenhaber bracket on the Hochschild cohomology of its Koszul dual algebra and hence on that of $A$ itself. Relations with some other brackets in literature are also discussed and several examples are given in detail.
Algebraic geometry
Associative rings and algebras
111
160
10.4171/JNCG/11-1-4
http://www.ems-ph.org/doi/10.4171/JNCG/11-1-4
Étale twists in noncommutative algebraic geometry and the twisted Brauer space
Benjamin
Antieau
University of Illinois at Chicago, CHICAGO, UNITED STATES
Derived categories, twisted forms, Hochschild cohomology, and Brauer groups
This paper studies étale twists of derived categories of schemes and associative algebras. A general method, based on a new construction called the twisted Brauer space, is given for classifying étale twists, and a complete classification is carried out for genus 0 curves, quadrics, and noncommutative projective spaces. A partial classification is given for curves of higher genus. The techniques build upon my recent work with David Gepner on the Brauer groups of commutative ring spectra.
Algebraic geometry
Associative rings and algebras
Category theory; homological algebra
161
192
10.4171/JNCG/11-1-5
http://www.ems-ph.org/doi/10.4171/JNCG/11-1-5
Divided differences in noncommutative geometry: Rearrangement Lemma, functional calculus and expansional formula
Matthias
Lesch
Universität Bonn, BONN, GERMANY
Rearrangement lemma, divided difference, expansional formula, modular curvature, topological tensor product, Banach algebra, functional calculus, spectral measure, asymptotic expansion, noncommutative global analysis
We state a generalization of the Connes–Tretkoff–Moscovici Rearrangement Lemma and give a surprisingly simple (almost trivial) proof of it. Secondly, we put on a firm ground the multivariable functional calculus used implicitly in the Rearrangement Lemma and elsewhere in the recent modular curvature paper by Connes and Moscovici [3]. Furthermore, we show that the fantastic formulas connecting the one and two variable modular functions of loc. cit. are just examples of the plenty recursion formulas which can be derived from the calculus of divided differences. We show that the functions derived from the main integral occurring in the Rearrangement Lemma can be expressed in terms of divided differences of the Logarithm, generalizing the “modified Logarithm” of Connes–Tretkoff [4]. Finally, we show that several expansion formulas related to the Magnus expansion [13] have a conceptual explanation in terms of a multivariable functional calculus applied to divided differences.
Functional analysis
Operator theory
Global analysis, analysis on manifolds
Numerical analysis
193
223
10.4171/JNCG/11-1-6
http://www.ems-ph.org/doi/10.4171/JNCG/11-1-6
Functoriality of equivariant eta forms
Bo
Liu
Humboldt-Universität zu Berlin, BERLIN, GERMANY
Equivariant eta form, index theory and fixed point theory, Chern-Simons form
In this paper, we define the equivariant eta form of Bismut–Cheeger for a compact Lie group and establish a formula about the functoriality of equivariant eta forms with respect to the composition of two submersions, which is motivated by constructing the geometric model of equivariant differential K-theory.
Global analysis, analysis on manifolds
$K$-theory
225
307
10.4171/JNCG/11-1-7
http://www.ems-ph.org/doi/10.4171/JNCG/11-1-7
From homotopy operads to infinity-operads
Brice
Le Grignou
Université de Nice Sophia Antipolis, NICE CEDEX 2, FRANCE
Operads, higher categories, nerve functor
The goal of the present paper is to compare, in a precise way, two approaches of operads up to homotopy which appear in the literature. Namely, we construct a functor from the category of strict unital homotopy colored operads to the category of infinity-operads. The former notion, that we make precise, is the operadic generalization of the notion of A-infinity-categories and the latter notion was defined by Moerdijk–Weiss in order to generalize the simplicial notion of infinity-category of Joyal–Lurie. This functor extends in two directions the simplicial nerve of Faonte–Lurie for A-infinity-categories and the homotopy coherent nerve of Moerdijk–Weiss for differential graded operads; it is also shown to be equivalent to a big nerve à la Lurie for differential graded operads. We prove that it satisfies some homotopy properties with respect to weak equivalences and fibrations; for instance, it is shown to be a right Quillen functor.
Category theory; homological algebra
Algebraic topology
309
365
10.4171/JNCG/11-1-8
http://www.ems-ph.org/doi/10.4171/JNCG/11-1-8
Graph products of operator algebras
Martijn
Caspers
Utrecht University, UTRECHT, NETHERLANDS
Pierre
Fima
Université Denis Diderot (Paris 7), PARIS CEDEX 13, FRANCE
Graph products, von Neumann algebras, quantum groups, free products, approximation properties
Graph products for groups were defined by Green in her thesis [25] as a generalization of both Cartesian and free products. In this paper we define the corresponding graph product for reduced and maximal $C^*$ -algebras, von Neumann algebras and quantum groups. We prove stability properties including permanence of II$_1$-factors, the Haagerup property, exactness and, under suitable conditions, the property of rapid decay for quantum groups.
Functional analysis
Group theory and generalizations
367
411
10.4171/JNCG/11-1-9
http://www.ems-ph.org/doi/10.4171/JNCG/11-1-9
2
Smooth geometry of the noncommutative pillow, cones and lens spaces
Tomasz
Brzeziński
Swansea University, UK and University of Bialystok, Poland
Andrzej
Sitarz
Jagiellonian University, Kraków, and Polish Academy of Sciences, Warsaw, Poland
Integrable differential calculus, Dirac operator, noncommutative pillow, quantum cone, quantum lens space
This paper proposes a new notion of smoothness of algebras, termed differential smoothness, that combines the existence of a top form in a differential calculus over an algebra together with a strong version of the Poincaré duality realized as an isomorphism between complexes of differential and integral forms. The quantum two- and three-spheres, disc, plane and the noncommutative torus are all smooth in this sense. Noncommutative coordinate algebras of deformations of several examples of classical orbifolds such as the pillow orbifold, singular cones and lens spaces are also differentially smooth. Although surprising this is not fully unexpected as these algebras are known to be homologically smooth. The study of Riemannian aspects of the noncommutative pillow and Moyal deformations of cones leads to spectral triples that satisfy the orientability condition that is known to be broken for classical orbifolds.
Global analysis, analysis on manifolds
413
449
10.4171/JNCG/11-2-1
http://www.ems-ph.org/doi/10.4171/JNCG/11-2-1
The Novikov conjecture on Cheeger spaces
Pierre
Albin
University of Illinois at Urbana-Champaign, USA
Eric
Leichtnam
Institut de Mathématiques de Jussieu-Paris Rive Gauche, France
Rafe
Mazzeo
Stanford University, USA
Paolo
Piazza
Università di Roma La Sapienza, Italy
Stratified spaces, $L^2$-cohomology, ideal boundary conditions, Cheeger spaces, higher signatures, stratified homotopy invariance, K-theory, higher index theory
We prove the Novikov conjecture on oriented Cheeger spaces whose fundamental group satisfies the strong Novikov conjecture. A Cheeger space is a stratified pseudomanifold admitting, through a choice of ideal boundary conditions, an $L^2$-de Rham cohomology theory satisfying Poincaré duality. We prove that this cohomology theory is invariant under stratified homotopy equivalences and that its signature is invariant under Cheeger space cobordism. Analogous results, after coupling with a Mischenko bundle associated to any Galois covering, allow us to carry out the analytic approach to the Novikov conjecture: we define higher analytic signatures of a Cheeger space and prove that they are stratified homotopy invariants whenever the assembly map is rationally injective. Finally we show that the analytic signature of a Cheeger space coincides with its topological signature as defined by Banagl.
Functional analysis
$K$-theory
Several complex variables and analytic spaces
Global analysis, analysis on manifolds
451
506
10.4171/JNCG/11-2-2
http://www.ems-ph.org/doi/10.4171/JNCG/11-2-2
Riemannian curvature of the noncommutative 3-sphere
Joakim
Arnlind
Linköping University, Sweden
Mitsuru
Wilson
University of Western Ontario, London, Canada
Noncommutative geometry, 3-sphere, Riemannian curvature, Levi-Civita connection
In order to investigate to what extent the calculus of classical (pseudo-)Riemannian manifolds can be extended to a noncommutative setting, we introduce pseudo-Riemannian calculi of modules over noncommutative algebras. In this framework, it is possible to prove an analogue of Levi-Civita’s theorem, which states that there exists at most one torsion-free and metric connection for a given (metric) module, satisfying the requirements of a real metric calculus. Furthermore, the corresponding curvature operator has the same symmetry properties as the classical Riemannian curvature. As our main motivating example, we consider a pseudo-Riemannian calculus over the noncommutative 3-sphere and explicitly determine the torsion-free and metric connection, as well as the curvature operator together with its scalar curvature.
Functional analysis
Global analysis, analysis on manifolds
507
536
10.4171/JNCG/11-2-3
http://www.ems-ph.org/doi/10.4171/JNCG/11-2-3
The Ext algebra of a Brauer graph algebra
Edward
Green
Virginia Tech, Blacksburg, USA
Sibylle
Schroll
University of Leicester, UK
Nicole
Snashall
University of Leicester, UK
Rachel
Taillefer
Université Clermont Auvergne, Université Blaise Pascal, Clermont-Ferrand, France
Brauer graph algebra, Koszul, $d$-Koszul, Ext algebra, finite generation
In this paper we study finite generation of the Ext algebra of a Brauer graph algebra by determining the degrees of the generators. As a consequence we characterize the Brauer graph algebras that are Koszul and those that are ${\mathcal K}_2$.
Associative rings and algebras
537
579
10.4171/JNCG/11-2-4
http://www.ems-ph.org/doi/10.4171/JNCG/11-2-4
The Gauss–Manin connection for the cyclic homology of smooth deformations, and noncommutative tori
Allan
Yashinski
University of Hawaii at Manoa, Honolulu, USA
Gauss–Manin connection, smooth deformation, cyclic homology, cyclic cohomology, K-theory, Chern–Connes pairing, Banach algebra, weak bidimension, noncommutative tori
Given a smooth deformation of topological algebras, we define Getzler’s Gauss–Manin connection on the periodic cyclic homology of the corresponding smooth field of algebras. Basic properties are investigated including the interaction with the Chern–Connes pairing with $K$-theory. We use the Gauss–Manin connection to prove a rigidity result for periodic cyclic cohomology of Banach algebras with finite weak bidimension. Then we illustrate the Gauss–Manin connection for the deformation of noncommutative tori. We use the Gauss–Manin connection to identify the periodic cyclic homology of a noncommutative torus with that of the commutative torus via a parallel translation isomorphism.We explicitly calculate the parallel translation maps and use them to describe the behavior of the Chern–Connes pairing under this deformation.
Functional analysis
Associative rings and algebras
581
639
10.4171/JNCG/11-2-5
http://www.ems-ph.org/doi/10.4171/JNCG/11-2-5
Part II, Free actions of compact groups on C*-algebras
Kay
Schwieger
University of Helsinki, Finland
Stefan
Wagner
Universität Hamburg, Germany
Weakly cleft action, C*-algebra, factor system, cocycle action
We study a simple class of free actions of non-Abelian groups on unital C *-algebras, namely cleft actions. These are characterized by the fact that the associated noncommutative vector bundles are trivial. In particular, we provide a complete classification theory for these actions and describe its relations to classical principal bundles.
Functional analysis
Associative rings and algebras
Dynamical systems and ergodic theory
Algebraic topology
641
668
10.4171/JNCG/11-2-6
http://www.ems-ph.org/doi/10.4171/JNCG/11-2-6
Spectral triples from bimodule connections and Chern connections
Edwin
Beggs
Swansea University, UK
Shahn
Majid
Queen Mary University of London, UK
Dirac operator, noncommutative differential calculus, bimodule connections, Chern connections, spectral triple, quantum sphere
We give a geometrical construction of Connes spectral triples or noncommutative Dirac operators $\slashed D$ starting with a bimodule connection on the proposed spinor bundle. The theory is applied to the example of $M_2(\mathbb C)$, and also applies to the standard $q$-sphere and the $q$-disk with the right classical limit and all properties holding except for $\mathcal J$ now being a twisted isometry. We also describe a noncommutative Chern construction from holomorphic bundles which in the $q$-sphere case provides the relevant bimodule connection.
Global analysis, analysis on manifolds
Functional analysis
Quantum theory
669
701
10.4171/JNCG/11-2-7
http://www.ems-ph.org/doi/10.4171/JNCG/11-2-7
Factorization homology and calculus à la Kontsevich Soibelman
Geoffroy
Horel
Universität Münster, Germany
Factorization homology, non-commutative calculus, Hochschild (co)homology, little disk operad, Swiss-cheese operad
We use factorization homology over manifolds with boundaries in order to construct operations on Hochschild cohomology and Hochschild homology. These operations are parametrized by a colored operad involving disks on the surface of a cylinder defined by Kontsevich and Soibelman. The formalism of the proof extends without difficulties to a higher dimensional situation. More precisely, we can replace associative algebras by algebras over the little disks operad of any dimensions, Hochschild homology by factorization (also called topological chiral) homology and Hochschild cohomology by higher Hochschild cohomology. Our result works in categories of chain complexes but also in categories of modules over a commutative ring spectrum giving interesting operations on topological Hochschild homology and cohomology.
Associative rings and algebras
Category theory; homological algebra
Algebraic topology
703
740
10.4171/JNCG/11-2-8
http://www.ems-ph.org/doi/10.4171/JNCG/11-2-8
Deformation quantization of integrable systems
Georgy
Sharygin
Lomonosov Moscow State University and ITEP, Moscow, Russia
Dmitry
Talalaev
Lomonosov Moscow State University and ITEP, Moscow, Russia
Quantization, integrable systems, Hochschild relative cohomology
In this paper we address the following question: is it always possible to choose a deformation quantization of a Poisson algebra $\mathcal{A}$ so that certain Poisson-commutative subalgebra $\mathcal{C}$ in it remains commutative? We define a series of cohomological obstructions to this, that take values in the Hochschild cohomology of $\mathcal{C}$ with coefficients in $\mathcal{A}$. In some particular case of the pair $(\mathcal{A},\mathcal{C})$ we reduce these classes to the classes of the Poisson relative cohomology of the Hochschild cohomology. We show, that in the case, when the algebra $\mathcal{C}$ is polynomial, these obstructions coincide with the previously known ones, those which were defined by Garay and van Straten.
Differential geometry
Functional analysis
741
756
10.4171/JNCG/11-2-9
http://www.ems-ph.org/doi/10.4171/JNCG/11-2-9
About the convolution of distributions on groupoids
Jean-Marie
Lescure
Université Blaise Pascal, Aubière, France
Dominique
Manchon
Université Blaise Pascal, Aubière, France
Stéphane
Vassout
Institut de Mathématiques de Jussieu - Paris Rive Gauche, France
Convolution of distributions, Lie groupoids, wave front set
We review the properties of transversality of distributions with respect to submersions. This allows us to construct a convolution product for a large class of distributions on Lie groupoids. We get a unital involutive algebra $\mathcal E_{r,s}'(G,\Omega^{1/2})$ enlarging the convolution algebra $C^\infty_c(G,\Omega^{1/2})$ associated with any Lie groupoid $G$. We prove that $G$-operators are convolution operators by transversal distributions. We also investigate the microlocal aspects of the convolution product. We give sufficient conditions on wave front sets to compute the convolution product and we show that the wave front set of the convolution product of two distributions is essentially the product of their wave front sets in the symplectic groupoid $T^*G$ of Coste–Dazord–Weinstein. This also leads to a subalgebra $\mathcal E_{a}'(G,\Omega^{1/2})$ of $\mathcal E_{r,s}'(G,\Omega^{1/2})$ which contains for instance the algebra of pseudodifferential $G$-operators and a class of Fourier integral $G$-operators which will be the central theme of a forthcoming paper.
Functional analysis
Topological groups, Lie groups
Global analysis, analysis on manifolds
757
789
10.4171/JNCG/11-2-10
http://www.ems-ph.org/doi/10.4171/JNCG/11-2-10
Finiteness and paradoxical decompositions in C*-dynamical systems
Timothy
Rainone
University of Waterloo, Canada
C*-algebras, crossed products, K-theory, dynamical systems
We discuss the interplay between $K$-theoretical dynamics and the structure theory of certain \cstar-algebras arising from crossed products. In the presence of sufficiently many projections we associate to each noncommutative \cstar-system $(A,G,\alpha)$ a type semigroup $S(A,G,\alpha)$ which reflects much of the spirit of the underlying action. We characterize purely infinite as well as stably finite crossed products in terms of finiteness and infiniteness in the type semigroup. We explore the dichotomy between stable finiteness and pure infiniteness in certain classes of reduced crossed products by means of paradoxical decompositions.
Functional analysis
Operator theory
791
822
10.4171/JNCG/11-2-11
http://www.ems-ph.org/doi/10.4171/JNCG/11-2-11
3
A note on the Lichnerowicz vanishing theorem for proper actions
Weiping
Zhang
Nankai University, Tianjin, China
Positive scalar curvature, proper actions, vanishing theorem
We prove a Lichnerowicz type vanishing theorem for non-compact spin manifolds admiting proper cocompact actions. This extends a previous result of Ziran Liu who proves it for the case where the acting group is unimodular.
Global analysis, analysis on manifolds
823
826
10.4171/JNCG/11-3-1
http://www.ems-ph.org/doi/10.4171/JNCG/11-3-1
Cohomology of $\mathcal A_θ^\mathrm {alg} \rtimes \mathbb Z_2$ and its Chern–Connes pairing
Safdar
Quddus
National Institute of Science Education & Research, Jatni, India
Cyclic cohomology, noncommutative torus, Chern–Connes index
We calculate the Hochschild and cyclic cohomology of the noncommutative $\mathbb Z_2$ toroidal algebraic orbifold $\mathcal A_θ^\mathrm {alg} \rtimes \mathbb Z_2$. We also calculate the Chern–Connes pairing of the even periodic cyclic cocycles with the known elements of $K_0 (\mathcal A_θ^\mathrm {alg} \rtimes \mathbb Z_2)$.
Global analysis, analysis on manifolds
Category theory; homological algebra
827
843
10.4171/JNCG/11-3-2
http://www.ems-ph.org/doi/10.4171/JNCG/11-3-2
The representation theory of non-commutative $\mathcal O$(GL$_2)$
Theo
Raedschelders
Free University of Brussels, Belgium
Michel
Van den Bergh
University of Hasselt, Diepenbeek, Belgium
Hopf algebras, monoidal categories, quasi-hereditary algebras
In our companion paper "The Manin Hopf algebra of a Koszul Artin–Schelter regular algebra is quasi-hereditary" we used the Tannaka–Krein formalism to study the universal coacting Hopf algebra $\underline {\mathrm {aut}}(A)$ for a Koszul Artin–Schelter regular algebra $A$. In this paper we study in detail the case $A=k[x,y]$. In particular we give a more precise description of the standard and costandard representations of $\underline {\mathrm {aut}}(A)$ as a coalgebra and we show that the latter can be obtained by induction from a Borel quotient algebra. Finally we give a combinatorial characterization of the simple $\underline {\mathrm {aut}}(A)$-representations as tensor products of $\underline {\mathrm {end}}(A)$-representations and their duals.
Associative rings and algebras
Group theory and generalizations
845
885
10.4171/JNCG/11-3-3
http://www.ems-ph.org/doi/10.4171/JNCG/11-3-3
Boundary value problems with Atiyah–Patodi–Singer type conditions and spectral triples
Ubertino
Battisti
Università di Torino, Italy
Joerg
Seiler
Università di Torino, Italy
Spectral triples, manifolds with boundary, boundary value problems with APS-type conditions, pseudodifferential operators
We study realizations of pseudodifferential operators acting on sections of vector-bundles on a smooth, compact manifold with boundary, subject to conditions of Atiyah–Patodi–Singer type. Ellipticity and Fredholm property, compositions, adjoints and self-adjointness of such realizations are discussed. We construct regular spectral triples $(\mathcal {A,H,D})$ for manifolds with boundary of arbitrary dimension, where $\mathcal H$ is the space of square integrable sections. Starting out from Dirac operators with APS-conditions, these triples are even in case of even dimensional manifolds; we show that the closure of $\mathcal A$ in $\mathcal L(\mathcal H)$ coincides with the continuous functions on the manifold being constant on each connected component of the boundary.
Global analysis, analysis on manifolds
Partial differential equations
Operator theory
887
917
10.4171/JNCG/11-3-4
http://www.ems-ph.org/doi/10.4171/JNCG/11-3-4
Orders of Nikshych's Hopf algebra
Juan
Cuadra
University of Almeria, Spain
Ehud
Meir
Universität Hamburg, Germany
Fusion categories, semisimple Hopf algebras, Hopf orders, group schemes, cyclotomic integers
Let $p$ be an odd prime number and $K$ a number field having a primitive $p$th root of unity $\zeta_p$. We prove that Nikshych's non group-theoretical Hopf algebra $H_p$, which is defined over $\mathbb Q(\zeta_p)$, admits a Hopf order over the ring of integers $\mathcal O_K$ if and only if there is an ideal $I$ of $\mathcal O_K$ such that $I^{2(p-1)} = (p)$. This condition does not hold in a cyclotomic field. Hence this gives an example of a semisimple Hopf algebra over a number field not admitting a Hopf order over any cyclotomic ring of integers. Moreover, we show that, when a Hopf order over $\mathcal O_K$ exists, it is unique and we describe it explicitly.
Associative rings and algebras
Algebraic geometry
Category theory; homological algebra
919
955
10.4171/JNCG/11-3-5
http://www.ems-ph.org/doi/10.4171/JNCG/11-3-5
$\mathcal A_{\infty}$-functors and homotopy theory of dg-categories
Giovanni
Faonte
Yale University, New Haven, USA
$\mathcal A_{\infty}$ functor, dg-category, $(\infty, 2)$-category, simplicial localization
In this paper we prove that Töen’s derived enrichment of the model category of dg-categories defined by Tabuada, is computed by the dg-category of $\mathcal A_{\infty}$-functors. This approach was suggested by Kontsevich. We further put this construction into the framework of $(\infty, 2)$-categories. Namely, we enhance the categories dgCat and $\mathcal A_{\infty}$ Cat, of dg and $\mathcal A_{\infty}$-categories, to $(\infty, 2)$-categories using the nerve construction of [4] and the $\mathcal A_{\infty}$-formalism. We prove that the $(\infty, 1)$-truncation of to the $(\infty, 2)$-category of dg-categories is a model for the simplicial localization at the model structure of Tabuada. As an application, we prove that the homotopy groups of the mapping space of endomorphisms at the identity functor in the $(\infty, 2)$-category of $\mathcal A_{\infty}$-categories compute the Hochschild cohomology.
Category theory; homological algebra
957
1000
10.4171/JNCG/11-3-6
http://www.ems-ph.org/doi/10.4171/JNCG/11-3-6
Langlands functorality in $K$-theory for $C^*$-algebras. I. Base change
Kuok Fai
Chao
Shanghai University, China
Hang
Wang
University of Adelaide, Australia and East China Normal University, Shanghai, China
$K$-theory, local Langlands correspondence, base change, reduced group $C^*$-algebra, tempered representation
We compare representations of the real and complex general linear groups and special linear groups in the framework of $K$-theory, using base change on $L$-parameters. We introduce a notion of base change on $K$-theory involving the fixed point set of the reduced dual of a complex group. For general linear groups, we prove that the base change map is compatible with the Connes–Kasparov isomorphism.
Topological groups, Lie groups
Group theory and generalizations
Functional analysis
1001
1036
10.4171/JNCG/11-3-7
http://www.ems-ph.org/doi/10.4171/JNCG/11-3-7
Differentiable absorption of Hilbert $C^*$-modules, connections, and lifts of unbounded operators
Jens
Kaad
The University of Southern Denmark, Odense, Denmark
Hilbert $C*$-modules, derivations, differentiable absorption, Grassmann connections, regular unbounded operators
The Kasparov absorption (or stabilization) theorem states that any countably generated Hilbert $C*$-module is isomorphic to a direct summand in the standard module of square summable sequences in the base $C*$-algebra. In this paper, this result will be generalized by incorporating a densely defined derivation on the base $C*$-algebra. This leads to a differentiable version of the Kasparov absorption theorem. The extra compatibility assumptions needed are minimal: It will only be required that there exists a sequence of generators with mutual inner products in the domain of the derivation. The differentiable absorption theorem is then applied to construct densely defined connections (or correpondences) on Hilbert $C*$-modules. These connections can in turn be used to define selfadjoint and regular "lifts" of unbounded operators which act on an auxiliary Hilbert $C*$-module.
Functional analysis
Operator theory
Differential geometry
Global analysis, analysis on manifolds
1037
1068
10.4171/JNCG/11-3-8
http://www.ems-ph.org/doi/10.4171/JNCG/11-3-8
(Almost) C*-algebras as sheaves with self-action
Cecilia
Flori
University of Waikato, Hamilton, New Zealand and Perimeter Institute for Theoretical Physics, Waterloo, Canada
Tobias
Fritz
Max Planck Institute for Mathematics, Leipzig, Germany and Perimeter Institute for Theoretical Physics, Waterloo, Canada
Axiomatics of C*-algebras, sheaf theory, algebraic quantum mechanics, topos quantum theory
Via Gelfand duality, a unital C*-algebra $A$ induces a functor from compact Hausdorff spaces to sets, $\mathsf{CHaus}\to\mathsf{Set}$. We show how this functor encodes standard functional calculus in $A$ as well as its multivariate generalization. Certain sheaf conditions satisfied by this functor provide a further generalization of functional calculus. Considering such sheaves $\mathsf{CHaus}\to\mathsf{Set}$ abstractly, we prove that the piecewise C*-algebras of van den Berg and Heunen are equivalent to a full subcategory of the category of sheaves, where a simple additional constraint characterizes the objects in the subcategory. It is open whether this additional constraint holds automatically, in which case piecewise C*-algebras would be the same as sheaves $\mathsf{CHaus}\to\mathsf{Set}$. Intuitively, these structures capture the commutative aspects of C*-algebra theory. In order to find a complete reaxiomatization of unital C*-algebras within this language, we introduce almost C*-algebras as piecewise C*-algebras equipped with a notion of inner automorphisms in terms of a self-action. We provide some evidence for the conjecture that the forgetful functor from unital C*-algebras to almost C*-algebras is fully faithful, and ask whether it is an equivalence of categories. We also develop an analogous notion of \emph{almost group}, and prove that the forgetful functor from groups to almost groups is not full. In terms of quantum physics, our work can be seen as an attempt at a reconstruction of quantum theory from physically meaningful axioms, as realized by Hardy and others in a different framework. Our ideas are inspired by and also provide new input for the topos-theoretic approach to quantum theory.
Functional analysis
Category theory; homological algebra
Group theory and generalizations
1069
1113
10.4171/JNCG/11-3-9
http://www.ems-ph.org/doi/10.4171/JNCG/11-3-9
Motivic Donaldson–Thomas invariants of some quantized threefolds
Alberto
Cazzaniga
Stellenbosch University, South Africa
Andrew
Morrison
ETH Zürich, Switzerland
Brent
Pym
University of Edinburgh, UK
Balázs
Szendrői
University of Oxford, UK
Donaldson–Thomas theory, motivic vanishing cycle, Calabi–Yau algebra, quiver representation, dimensional reduction
This paper is motivated by the question of howmotivic Donaldson–Thomas invariants behave in families. We compute the invariants for some simple families of noncommutative Calabi–Yau threefolds, defined by quivers with homogeneous potentials. These families give deformation quantizations of affine three-space, the resolved conifold, and the resolution of the transversal $A_n$-singularity. It turns out that their invariants are generically constant, but jump at special values of the deformation parameter, such as roots of unity. The corresponding generating series are written in closed form, as plethystic exponentials of simple rational functions. While our results are limited by the standard dimensional reduction techniques that we employ, they nevertheless allow us to conjecture formulae for more interesting cases, such as the elliptic Sklyanin algebras.
Algebraic geometry
Associative rings and algebras
1115
1139
10.4171/JNCG/11-3-10
http://www.ems-ph.org/doi/10.4171/JNCG/11-3-10
A symbol calculus for foliations
Moulay-Tahar
Benameur
Université de Montpellier, France
James
Heitsch
University of Illinois at Chicago, USA
Foliations, asymptotic calculus, pseudodifferential operators, index theory, noncommutative geometry, global analysis
The classical Getzler rescaling theorem of [15] is extended to the transverse geometry of foliations. More precisely, a Getzler rescaling calculus, [15], as well as a Block–Fox calculus of asymptotic pseudodifferential operators (A$\Psi$DOs), [10], is constructed for all transversely spin foliations. This calculus applies to operators of degree $m$ globally times degree $\ell$ in the leaf directions, and is thus an appropriate tool for a better understanding of the index theory of transversely elliptic operators on foliations [13]. The main result is that the composition of A$\Psi$DOs is again an A$\Psi$DO, and includes a formula for the leading symbol. Our formula is more complicated due to its wide generality but its form is essentially the same, and it simplifies notably for Riemannian foliations. In short, we construct an asymptotic pseudodifferential calculus for the “leaf space” of any foliation. Applications will be derived in [5,6] where we give a Getzler-like proof of a local topological formula for the Connes–Chern character of the Connes–Moscovici spectral triple of [20], as well as the (semi-finite) spectral triple given in [5], yielding an extension of the seminal Atiyah–Singer $L^2$ covering index theorem, [2], to coverings of “leaf spaces” of foliations.
Operator theory
Differential geometry
1141
1194
10.4171/JNCG/11-3-11
http://www.ems-ph.org/doi/10.4171/JNCG/11-3-11
Crossed products for actions of crossed modules on C*-algebras
Alcides
Buss
Universidade Federal de Santa Catarina, Florianópolis, Brazil
Ralf
Meyer
Universität Göttingen, Germany
C*-algebra, crossed module, Fell bundle, Takesaki–Takai duality
We decompose the crossed product functor for actions of crossed modules of locally compact groups on C -algebras into more elementary constructions: taking crossed products by group actions and fibres in C -algebras over topological spaces. For this, we extend Takesaki–Takai duality to Abelian crossed modules; describe the crossed product for an extension of crossed modules; show that equivalent crossed modules have equivalent categories of actions on C -algebras; and show that certain crossed modules are automatically equivalent to Abelian crossed modules.
Functional analysis
Category theory; homological algebra
1195
1235
10.4171/JNCG/11-3-12
http://www.ems-ph.org/doi/10.4171/JNCG/11-3-12
4
Monodromy of the Gauss–Manin connection for deformation by group cocycles
Makoto
Yamashita
Ochanomizu University, Tokyo, Japan
Cyclic homology, group cohomology, deformation quantization
We consider the 2-cocycle deformation of algebras graded by discrete groups. The action of the Maurer–Cartan form on cyclic cohomology is shown to be cohomologous to the cup product action of the group cocycle. This allows us to compute the monodromy of the Gauss–Manin connection in the strict deformation setting.
$K$-theory
Functional analysis
1237
1265
10.4171/JNCG/11-4-1
http://www.ems-ph.org/doi/10.4171/JNCG/11-4-1
12
15
2017
Representability of cohomological functors over extension fields
Alice
Rizzardo
SISSA, Trieste, Italy
Representability, base extension, Fourier–Mukai
We generalize a result of Orlov and Van den Bergh on the representability of a cohomological functor $H:D^{b}_{\mathrm{Coh}}(X)\to \underline{\mathrm{mod}}_{L}$ to the case where $L$ is a field extension of the base field $k$ of the variety $X$, with trdeg$_k L\leq 1$ or $L$ purely transcendental of degree 2. This result can be applied to investigate the behavior of an exact functor $F:D^{b}_{\mathrm{Coh}}(X)\to D^{b}_{\mathrm{Coh}}(Y)$ with $X$ and $Y$ smooth projective varieties and dim $Y\leq 1$ or $Y$ a rational surface. We show that for any such $F$ there exists a "generic kernel" $A$ in $D^{b}_{\mathrm{Coh}}(X\times Y)$, such that $F$ is isomorphic to the Fourier–Mukai transform with kernel $A$ after composing both with the pullback to the generic point of $Y$.
Category theory; homological algebra
Algebraic geometry
1267
1287
10.4171/JNCG/11-4-2
http://www.ems-ph.org/doi/10.4171/JNCG/11-4-2
12
15
2017
Koszul pairs and applications
Pascual
Jara
Universidad de Granada, Spain
Javier
López Peña
University College London, UK
Dragoş
Ştefan
University of Bucharest, Romania
Koszul rings, Koszul pairs, Hochschild (co)homology, twisted tensor products
Let $R$ be a semisimple ring. A pair $(A,C)$ is called almost-Koszul if $A$ is a connected graded $R$-ring and $C$ is a compatible connected graded $R$-coring. To an almost-Koszul pair one associates three chain complexes and three cochain complexes such that one of them is exact if and only if the others are so. In this situation $(A,C)$ is said to be Koszul. One proves that a connected $R$-ring $A$ is Koszul if and only if there is a connected $R$-coring $C$ such that $(A,C)$ is Koszul. This result allows us to investigate the Hochschild (co)homology of Koszul rings. We apply our method to show that the twisted tensor product of two Koszul rings is Koszul. More examples and applications of Koszul pairs, including a generalization of Fröberg Theorem [12], are discussed in the last part of the paper.
Associative rings and algebras
1289
1350
10.4171/JNCG/11-4-3
http://www.ems-ph.org/doi/10.4171/JNCG/11-4-3
12
15
2017
Gerstenhaber brackets on Hochschild cohomology of twisted tensor products
Lauren
Grimley
Spring Hill College, Mobile, USA
Van
Nguyen
Hood College, Frederick, USA
Sarah
Witherspoon
Texas A&M University, College Station, USA
Hochschild cohomology, Gerstenhaber brackets, twisted tensor products, quantum complete intersections
We construct the Gerstenhaber bracket on Hochschild cohomology of a twisted tensor product of algebras, and, as examples, compute Gerstenhaber brackets for some quantum complete intersections arising in the work of Buchweitz, Green, Madsen, and Solberg.We prove that a subalgebra of the Hochschild cohomology ring of a twisted tensor product, on which the twisting is trivial, is isomorphic, as a Gerstenhaber algebra, to the tensor product of the respective subalgebras of the Hochschild cohomology rings of the factors.
Associative rings and algebras
1351
1379
10.4171/JNCG/11-4-4
http://www.ems-ph.org/doi/10.4171/JNCG/11-4-4
12
15
2017
Smooth crossed product of minimal unique ergodic diffeomorphism of odd sphere
Hongzhi
Liu
Jilin University, Changchun, China
Smooth crossed products, cyclic cohomology
For minimal unique ergodic diffeomorphisms $\alpha_n$ of $S^{2n+1} (n > 0)$ and $\alpha_m$ of $S^{2m+1}(m>0)$, the $C^*$-crossed product algebra $C(S^{2n+1})\rtimes_{\alpha_n} \mathbb{Z}$ is isomorphic to $C(S^{2m+1})\rtimes_{\alpha_m} \mathbb{Z}$ even though $n\neq m$. However, by cyclic cohomology, we show that smooth crossed product algebra $C^\infty(S^{2n+1})\rtimes_{\alpha_n} \mathbb{Z}$ is not isomorphic to $C^\infty(S^{2m+1})\rtimes_{\alpha_m} \mathbb{Z}$ if~$n\neq m$.
$K$-theory
Functional analysis
Global analysis, analysis on manifolds
1381
1393
10.4171/JNCG/11-4-5
http://www.ems-ph.org/doi/10.4171/JNCG/11-4-5
12
15
2017
The noncommutative Kalton–Peck spaces
Félix
Cabello Sánchez
Universidad de Extremadura, Badajoz, Spain
Noncommutative $L_p$-spaces, Kalton–Peck spaces, twisted sum, complex interpolation
For every von Neumann algebra $\mathcal M$ and $0
Functional analysis
Abstract harmonic analysis
1395
1412
10.4171/JNCG/11-4-6
http://www.ems-ph.org/doi/10.4171/JNCG/11-4-6
12
15
2017
Spectral triples for nested fractals
Daniele
Guido
Università di Roma Tor Vergata, Rome, Italy
Tommaso
Isola
Università di Roma Tor Vergata, Rome, Italy
Spectral triple, nested fractal, self-similar energy, Hausdorff dimension, noncommutative distance
It is shown that, for nested fractals [31], the main structural data, such as the Hausdorff dimension and measure, the geodesic distance (when it exists) induced by the immersion in $\mathbb R^n$, and the self-similar energy can all be recovered by the description of the fractals in terms of the spectral triples considered in [18].
Global analysis, analysis on manifolds
Measure and integration
1413
1436
10.4171/JNCG/11-4-7
http://www.ems-ph.org/doi/10.4171/JNCG/11-4-7
12
15
2017
Decorated Feynman categories
Ralph
Kaufmann
Purdue University, West Lafayette, USA and Max Planck Institut für Mathematik, Bonn, Germany
Jason
Lucas
Purdue University, West Lafayette, USA
Feynman categories, monoidal category, operads, moduli spaces, operads, non- Sigma operads, enrichment, modular operads, surfaces, functors
In [12], the new concept of Feynman categories was introduced to simplify the discussion of operad–like objects. In this present paper, we demonstrate the usefulness of this approach, by introducing the concept of decorated Feynman categories. The procedure takes a Feynman category $\mathfrak F$ and a functor $\mathcal O$ to a monoidal category to produce a new Feynman category $\mathfrak F_{\mathrm {dec}\mathcal O}$. This in one swat explains the existence of non–sigma operads, non–sigma cyclic operads, and the non–sigma–modular operads of Markl as well as all the usual candidates simply from the category $\mathfrak G$, which is a full subcategory of the category of graphs of [2]. Moreover, we explain the appearance of terminal objects noted in [16]. We can then easily extend this for instance to the dihedral case. Furthermore, we obtain graph complexes and all other known operadic type notions from decorating and restricting the basic Feynman category $\mathfrak G$ of aggregates of corollas. We additionally show that the construction is functorial. There are further geometric and number theoretic applications, which will follow in a separate preprint.
Algebraic topology
Algebraic geometry
Category theory; homological algebra
Several complex variables and analytic spaces
1437
1464
10.4171/JNCG/11-4-8
http://www.ems-ph.org/doi/10.4171/JNCG/11-4-8
12
15
2017
Ring-theoretic blowing down. I
Daniel
Rogalski
University of California San Diego, La Jolla, USA
Susan
Sierra
University of Edinburgh, UK
J. Toby
Stafford
The University of Manchester, UK
Noncommutative projective geometry, noncommutative surfaces, Sklyanin algebras, noetherian graded rings, noncommutative blowing up and blowing down, Castelnuovo’s contraction theorem
One of the major open problems in noncommutative algebraic geometry is the classification of noncommutative projective surfaces (or, slightly more generally, of noetherian connected graded domains of Gelfand–Kirillov dimension 3). Earlier work of the authors classified the connected graded noetherian subalgebras of Sklyanin algebras using a noncommutative analogue of blowing up. In order to understand other algebras birational to a Sklyanin algebra, one also needs a notion of blowing down. This is achieved in this paper, where we give a noncommutative analogue of Castelnuovo’s classic theorem that (–1)-lines on a smooth surface can be contracted. The resulting noncommutative blown-down algebra has pleasant properties; in particular it is always noetherian and is smooth if the original noncommutative surface is smooth. In a companion paper we will use this technique to construct explicit birational transformations between various noncommutative surfaces which contain an elliptic curve.
Algebraic geometry
Associative rings and algebras
Category theory; homological algebra
1465
1520
10.4171/JNCG/11-4-9
http://www.ems-ph.org/doi/10.4171/JNCG/11-4-9
12
15
2017
On compact bicrossed products
Pierre
Fima
Université Denis Diderot Paris 7, Paris, France
Kunal
Mukherjee
Indian Institute of Technology Madras, Chennai, India
Issan
Patri
The Institute of Mathematical Sciences, Chennai, India
Compact quantum group, bicrossed product, matched pair, property T, Haagerup property
We make a comprehensive and self-contained study of compact bicrossed products arising from matched pairs of discrete groups and compact groups. We exhibit an automatic regularity property of such a matched pair and describe the representation theory and the fusion rules of the associated bicrossed product $\mathbb G$. We investigate the relative co-property $(T)$ and the relative co-Haagerup property of the pair comprising of the compact group and the bicrossed product, discuss property $(T)$ and Haagerup property of the discrete dual $\widehat{\mathbb G}$, and review co-amenability of $\mathbb G$ as well. We distinguish two such non-trivial compact bicrossed products with relative co-property $(T)$ and also provide an infinite family of pairwise non isomorphic non-trivial discrete quantum groups with property $(T)$, the existence of even one of the latter was unknown. Finally, we examine all the properties mentioned above for the crossed product quantum group given by an action by quantum automorphisms of a discrete group on a compact quantum group, and also establish the permanence of rapid decay and weak amenability and provide several explicit examples.
Functional analysis
Quantum theory
1521
1591
10.4171/JNCG/11-4-10
http://www.ems-ph.org/doi/10.4171/JNCG/11-4-10
12
15
2017
Crossed products by compact group actions with the Rokhlin property
Eusebio
Gardella
Universität Münster, Germany
Rokhlin property, crossed product, fixed point algebra, weak semiprojectivity
We present a systematic study of the structure of crossed products and fixed point algebras by compact group actions with the Rokhlin property on not necessarily unital $C^*$-algebras. Our main technical result is the existence of an approximate homomorphism from the algebra to its subalgebra of fixed points, which is a left inverse for the canonical inclusion. Upon combining this with results regarding local approximations, we show that a number of classes characterized by inductive limit decompositions with weakly semiprojective building blocks, are closed under formation of crossed products by such actions. Similarly, in the presence of the Rokhlin property, if the algebra has any of the following properties, then so do the crossed product and the fixed point algebra: being a Kirchberg algebra, being simple and having tracial rank zero or one, having real rank zero, having stable rank one, absorbing a strongly self-absorbing $C^*$-algebra, satisfying the Universal Coefficient Theorem (in the simple, nuclear case), and being weakly semiprojective. The ideal structure of crossed products and fixed point algebras by Rokhlin actions is also studied. The methods of this paper unify, under a single conceptual approach, the work of a number of authors, who used rather different techniques. Our methods yield new results even in the well-studied case of finite group actions with the Rokhlin property.
Functional analysis
1593
1626
10.4171/JNCG/11-4-11
http://www.ems-ph.org/doi/10.4171/JNCG/11-4-11
12
15
2017
$\mathbb A^1$-homotopy invariants of corner skew Laurent polynomial algebras
Gonçalo
Tabuada
MIT, Cambridge, USA and Universidade Nova de Lisboa, Portugal
Corner skew Laurent polynomial algebra, Leavitt path algebra, algebraic $K$-theory, noncommutative mixed motives, noncommutative algebraic geometry
In this note we prove some structural properties of all the $\mathbb A^1$-homotopy invariants of corner skew Laurent polynomial algebras. As an application, we compute the mod-$l$ algebraic $K$-theory of Leavitt path algebras using solely the kernel/cokernel of the incidence matrix. This leads naturally to some vanishing and divisibility properties of the $K$-theory of these algebras.
Algebraic geometry
Associative rings and algebras
$K$-theory
1627
1643
10.4171/JNCG/11-4-12
http://www.ems-ph.org/doi/10.4171/JNCG/11-4-12
12
15
2017