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  • The index of $G$-transversally elliptic families. I
    J. Noncommut. Geom. (IF 0.727) Pub Date : 2020-11-03
    Alexandre Baldare

    We define and study the index map for families of $G$-transversally elliptic operators and introduce the multiplicity for a given irreducible representation as a virtual bundle over the base of the fibration. We then prove the usual axiomatic properties for the index map extending the Atiyah–Singer results [1]. Finally, we compute the Kasparov intersection product of our index class against the K-homology

    更新日期:2020-11-04
  • The index of $G$-transversally elliptic families. II
    J. Noncommut. Geom. (IF 0.727) Pub Date : 2020-11-03
    Alexandre Baldare

    We define the Chern character of the index class of a $G$-invariant family of $G$-transversally elliptic operators, see [6]. Next we study the Berline–Vergne formula for families in the elliptic and transversally elliptic case.

    更新日期:2020-11-04
  • The relative Mishchenko–Fomenko higher index and almost flat bundles. I. The relative Mishchenko–Fomenko index
    J. Noncommut. Geom. (IF 0.727) Pub Date : 2020-11-03
    Yosuke Kubota

    In this paper, the first of two, we introduce an alternative definition of the Chang–Weinberger–Yu relative higher index, which is thought of as a relative analogue of the Mishchenko–Fomenko index pairing. A main result of this paper is that our map coincides with the existing relative higher index maps. We make use of this fact for understanding the relative higher index. First, we relate the relative

    更新日期:2020-11-04
  • Partition quantum spaces
    J. Noncommut. Geom. (IF 0.727) Pub Date : 2020-09-03
    Stefan Jung; Moritz Weber

    We propose a definition of partition quantum spaces. They are given by universal $C*$-algebras whose relations come from partitions of sets. We ask for the maximal compact matrix quantum group acting on them. We show how those fit into the setting of easy quantum groups: Our approach yields spaces these groups are acting on. In a way, our partition quantum spaces arise as the first d columns of easy

    更新日期:2020-11-03
  • Quantum function algebras from finite-dimensional Nichols algebras
    J. Noncommut. Geom. (IF 0.727) Pub Date : 2020-10-12
    Marco Andrés Farinati; Gastón Andrés García

    We describe how to find quantum determinants and antipode formulas from braided vector spaces using the FRT-construction and finite-dimensional Nichols algebras. It improves the construction of quantum function algebras using quantum grassmanian algebras. Given a finite-dimensional Nichols algebra $\mathfrak B$, our method provides a Hopf algebra $H$ such that $\mathfrak B$ is a braided Hopf algebra

    更新日期:2020-11-03
  • The homology of the Katsura–Exel–Pardo groupoid
    J. Noncommut. Geom. (IF 0.727) Pub Date : 2020-10-21
    Eduard Ortega

    We compute the homology of the groupoid associated to the Katsura algebras, and show that they capture the $K$-theory of the $C^*$-algebras in the sense of the (HK) conjecture posted by Matui. Moreover, we show that several classifiable simple $C^*$-algebras are groupoid $C^*$-algebras of this class.

    更新日期:2020-11-03
  • On the graded algebras associated with Hecke symmetries
    J. Noncommut. Geom. (IF 0.727) Pub Date : 2020-10-12
    Serge Skryabin

    A Hecke symmetry $R$ on a finite dimensional vector space $V$ gives rise to two graded factor algebras $\mathbb S (V, R)$ and $\Lambda (V, R)$ of the tensor algebra of $V$ which are regarded as quantum analogs of the symmetric and the exterior algebras. Another graded algebra associated with $R$ is the Faddeev–Reshetikhin–Takhtajan bialgebra $A(R)$ which coacts on $\mathbb S (V, R)$ and $\Lambda (V

    更新日期:2020-11-03
  • The intertwiner spaces of non-easy group-theoretical quantum groups
    J. Noncommut. Geom. (IF 0.727) Pub Date : 2020-10-21
    Laura Maaßen

    In 2015, Raum and Weber gave a definition of group-theoretical quantum groups, a class of compact matrix quantum groups with a certain presentation as semi-direct product quantum groups, and studied the case of easy quantum groups. In this article we determine the intertwiner spaces of non-easy group-theoretical quantum groups. We generalise group-theoretical categories of partitions and use a fiber

    更新日期:2020-11-03
  • Deformation-obstruction theory for diagrams of algebras and applications to geometry
    J. Noncommut. Geom. (IF 0.727) Pub Date : 2020-10-15
    Severin Barmeier; Yaël Frégier

    Let $X$ be an algebraic variety over an algebraically closed field of characteristic $0$ and let $\Coh (X)$ denote its Abelian category of coherent sheaves. By the work of W. Lowen and M. Van den Bergh, it is known that the deformation theory of Coh($X$) as an Abelian category can be seen to be controlled by the Gerstenhaber–Schack complex associated to the restriction of the structure sheaf $\mathcal

    更新日期:2020-11-03
  • Quantization of a Poisson structure on products of principal affine spaces
    J. Noncommut. Geom. (IF 0.727) Pub Date : 2020-10-12
    Victor Mouquin

    We give the analogue for Hopf algebras of the polyuble Lie bialgebra construction by Fock and Rosli. By applying this construction to the Drinfeld–Jimbo quantum group, we obtain a deformation quantization $\mathbb C_\hslash[(N \backslash G)^m]$ of a Poisson structure $\pi^{(m)}$ on products $(N \backslash G)^m$ of principal affine spaces of a connected and simply connected complex semisimple Lie group

    更新日期:2020-11-03
  • Symmetries of slice monogenic functions
    J. Noncommut. Geom. (IF 0.727) Pub Date : 2020-10-21
    Fabrizio Colombo; Rolf Sören Kraußhar; Irene Sabadini

    In this paper we consider the symmetry behavior of slice monogenic functions under Möbius transformations. We describe the group under which slice monogenic functions are taken into slice monogenic functions. We prove a transformation formula for composing slice monogenic functions with Möbius transformations and describe their conformal invariance. Finally, we explain two construction methods to obtain

    更新日期:2020-11-03
  • The factor type of dissipative KMS weights on graph $C$*-algebras
    J. Noncommut. Geom. (IF 0.727) Pub Date : 2020-10-15
    Klaus Thomsen

    We calculate the $S$-invariant of Connes for the von Neumann algebra factors arising from KMS weights of a generalized gauge action on a simple graph $C$*-algebra when the associated measure on the infinite path space of the graph is dissipative under the action of the shift.

    更新日期:2020-11-03
  • Hochschild cohomology and orbifold Jacobian algebras associated to invertible polynomials
    J. Noncommut. Geom. (IF 0.727) Pub Date : 2020-08-12
    Alexey Basalaev; Atsushi Takahashi

    Let $f$ be an invertible polynomial and $G$ a group of diagonal symmetries of $f$. This note shows that the orbifold Jacobian algebra Jac$(f,G)$ of $(f,G)$ defined by [2] is isomorphic as a $\mathbb Z/2\mathbb ZZ$-graded algebra to the Hochschild cohomology $\mathsf{HH}^*(\mathrm {MF}_G(f))$ of the dg-category $\mathrm {MF}_G(f)$ of $G$-equivariant matrix factorizations of $f$ by calculating the product

    更新日期:2020-08-12
  • Bounds for the rank of the finite part of operator $K$-theory
    J. Noncommut. Geom. (IF 0.727) Pub Date : 2020-06-30
    Süleyman Kağan Samurkaş

    We derive a lower and an upper bound for the rank of the finite part of operator $K$-theory groups of maximal and reduced $C^*$-algebras of finitely generated groups. The lower bound is based on the amount of polynomially growing conjugacy classes of finite order elements in the group. The upper bound is based on the amount of torsion elements in the group. We use the lower bound to give lower bounds

    更新日期:2020-07-30
  • Additivity of higher rho invariant for the topological structure group from a differential point of view
    J. Noncommut. Geom. (IF 0.727) Pub Date : 2020-05-28
    Baojie Jiang; Hongzhi Liu

    In [16], Weinberger, Xie, and Yu proved that higher rho invariant associated to homotopy equivalence defines a group homomorphism from the topological structure group to the analytic structure group, $K$-theory of certain geometric $C^*$-algebras, by piecewise-linear approach. In this paper, we adapt part of Weinberger, Xie, and Yu’s work, to give a differential geometry theoretic proof of the additivity

    更新日期:2020-07-30
  • Descent of Hilbert $C$*-modules
    J. Noncommut. Geom. (IF 0.727) Pub Date : 2020-06-25
    Tyrone Crisp

    Let $F$ be a right Hilbert $C$*-module over a $C$*-algebra $B$, and suppose that $F$ is equipped with a left action, by compact operators, of a second $C$*-algebra $A$. Tensor product with $F$ gives a functor from Hilbert $C$*-modules over $A$ to Hilbert $C$*-modules over $B$. We prove that under certain conditions (which are always satisfied if, for instance, $A$ is nuclear), the image of this functor

    更新日期:2020-07-30
  • $A_{\infty}$-coderivations and the Gerstenhaber bracket on Hochschild cohomology
    J. Noncommut. Geom. (IF 0.727) Pub Date : 2020-06-26
    Cris Negron; Yury Volkov; Sarah Witherspoon

    We show that Hochschild cohomology of an algebra over a field is a space of infinity coderivations on an arbitrary projective bimodule resolution of the algebra. The Gerstenhaber bracket is the graded commutator of infinity coderivations. We thus generalize, to an arbitrary resolution, Stasheff’s realization of the Gerstenhaber bracket on Hochschild cohomology as the graded commutator of coderivations

    更新日期:2020-07-30
  • Property (T), property (F) and residual finiteness for discrete quantum groups
    J. Noncommut. Geom. (IF 0.727) Pub Date : 2020-07-02
    Angshuman Bhattacharya; Michael Brannan; Alexandru Chirvasitu; Shuzhou Wang

    We investigate connections between various rigidity and softness properties for discrete quantum groups. After introducing a notion of residual finiteness, we show that it implies the Kirchberg factorization property for the discrete quantum group in question. We also prove the analogue of Kirchberg’s theorem, to the effect that conversely, the factorization property and property (T) jointly imply

    更新日期:2020-07-30
  • Tannaka duality for enhanced triangulated categories I: reconstruction
    J. Noncommut. Geom. (IF 0.727) Pub Date : 2020-07-13
    Jonathan P. Pridham

    We develop Tannaka duality theory for dg categories. To any dg functor from a dg category $\mathcal A$ to finite-dimensional complexes, we associate a dg coalgebra $C$ via a Hochschild homology construction. When the dg functor is faithful, this gives a quasi-equivalence between the derived dg categories of $\mathcal A$-modules and of $C$-comodules. When $\mathcal A$ is Morita fibrant (i.e. an idempotent-complete

    更新日期:2020-07-30
  • Bivariant K-theory of generalized Weyl algebras
    J. Noncommut. Geom. (IF 0.727) Pub Date : 2020-07-19
    Julio Gutiérrez; Christian Valqui

    We compute the isomorphism class in $\mathfrak{KK}^{\mathrm {alg}}$ of all noncommutative generalized Weyl algebras $A=\mathbb C[h](\sigma, P)$,where $\sigma(h)=qh+h_0$ is an automorphism of $\mathcal C[h]$, except when $q\neq 1$ is a root of unity. In particular, we compute the isomorphism class in $\mathfrak{KK}^{\mathrm {alg}}$ of the quantum Weyl algebra, the primitive factors $B_{\lambda}$ of

    更新日期:2020-07-30
  • The nodal cubic and quantum groups at roots of unity
    J. Noncommut. Geom. (IF 0.727) Pub Date : 2020-07-21
    Ulrich Krähmer; Manuel Martins

    In a recent article, the coordinate ring of the nodal cubic was given the structure of a quantum homogeneous space. Here the corresponding coalgebra Galois extension is expressed in terms of quantum groups at roots of unity, and is shown to be cleft. Furthermore, the minimal quotient extensions are determined.

    更新日期:2020-07-30
  • Almost commutative $Q$-algebras and derived brackets
    J. Noncommut. Geom. (IF 0.727) Pub Date : 2020-07-13
    Andrew J. Bruce

    We introduce the notion of almost commutative $Q$-algebras and demonstrate how the derived bracket formalism of Kosmann–Schwarzbach generalises to this setting. In particular, we construct ‘almost commutative Lie algebroids’ following Vaıntrob’s $Q$-manifold understanding of classical Lie algebroids. We show that the basic tenets of the theory of Lie algebroids carry over verbatim to the almost commutative

    更新日期:2020-07-30
  • A Chern–Weil formula for the Chern character of a perfect curved module
    J. Noncommut. Geom. (IF 0.727) Pub Date : 2020-07-29
    Michael K. Brown; Mark E. Walker

    Let $k$ be a field of characteristic 0 and $\mathcal{A}$ a curved $k$-algebra. We obtain a Chern–Weil-type formula for the Chern character of a perfect $\mathcal{A}$-module taking values in $HN^{II}_0(\mathcal{A})$, the negative cyclic homology of the second kind associated to $\mathcal{A}$, when $\mathcal{A}$ satisfies a certain smoothness condition.

    更新日期:2020-07-30
  • Embedding of the derived Brauer group into the secondary $K$-theory ring
    J. Noncommut. Geom. (IF 0.727) Pub Date : 2020-07-21
    Gonçalo Tabuada

    In this note, making use of the recent theory of noncommutative motives, we prove that the canonical map from the derived Brauer group to the secondary Grothendieck ring has the following injectivity properties: in the case of a regular integral quasi-compact quasi-separated scheme, it is injective; in the case of an integral normal Noetherian scheme with a single isolated singularity, it distinguishes

    更新日期:2020-07-30
  • On the twisted tensor product of small dg categories
    J. Noncommut. Geom. (IF 0.727) Pub Date : 2020-07-29
    Boris Shoikhet

    Given two small dg categories $C,D$, defined over a field, we introduce their (non-symmetric) twisted tensor product $C\sotimes D$. We show that $-\sotimes D$ is left adjoint to the functor $\mathcal {Coh}(D,-)$, where $\mathcal {Coh}(D,E)$ is the dg category of dg functors $D\to E$ and their coherent natural transformations. This adjunction holds in the category of small dg categories (not in the

    更新日期:2020-07-30
  • The Higson–Roe sequence for étale groupoids. I. Dual algebras and compatibility with the BC map
    J. Noncommut. Geom. (IF 0.727) Pub Date : 2020-05-14
    Moulay-Tahar Benameur; Indrava Roy

    We introduce the dual Roe algebras for proper étale groupoid actions and deduce the expected Higson–Roe short exact sequence. When the action is co-compact, we show that the Roe $C^*$-ideal of locally compact operators is Morita equivalent to the reduced $C^*$-algebra of our groupoid, and we further identify the boundary map of the associated periodic six-term exact sequence with the Baum–Connes map

    更新日期:2020-07-20
  • Non-commutative crepant resolutions for some toric singularities. II
    J. Noncommut. Geom. (IF 0.727) Pub Date : 2020-05-18
    Špela Špenko; Michel Van den Bergh

    Using the theory of dimer models Broomhead proved that every 3-dimensional Gorenstein affine toric variety Spec $R$ admits a toric non-commutative crepant resolution (NCCR). We give an alternative proof of this result by constructing a tilting bundle on a (stacky) crepant resolution of Spec $R$ using standard toric methods. Our proof does not use dimer models.

    更新日期:2020-07-20
  • Hecke operators in $KK$-theory and the $K$-homology of Bianchi groups
    J. Noncommut. Geom. (IF 0.727) Pub Date : 2020-05-14
    Bram Mesland; Mehmet Haluk Şengün

    Let $\Gamma$ be a torsion-free arithmetic group acting on its associated global symmetric space $X$. Assume that $X$ is of non-compact type and let $\Gamma$ act on the geodesic boundary $\partial X$ of $X$. Via general constructions in $KK$-theory, we endow the $K$-groups of the arithmetic manifold $X / \Gamma$, of the reduced group $C^*$-algebra $C^*_r(\Gamma)$ and of the boundary crossed product

    更新日期:2020-07-20
  • Differential calculus over double Lie algebroids
    J. Noncommut. Geom. (IF 0.727) Pub Date : 2020-05-25
    Sophie Chemla

    M. Van den Bergh [20] defined the notion of a double Lie algebroid and showed that a double quasi-Poisson algebra gives rise to a double Lie algebroid.We give new examples of double Lie algebroids and develop a differential calculus in that context. We recover the non commutative Karoubi–de Rham complex [7, 9] and the double Poisson–Lichnerowicz cohomology [16] as particular cases of our construction

    更新日期:2020-07-20
  • Annular representations of free product categories
    J. Noncommut. Geom. (IF 0.727) Pub Date : 2020-05-14
    Shamindra Kumar Ghosh; Corey Jones; B. Madhav Reddy

    We provide a description of the annular representation category of the free product of two rigid C*-tensor categories.

    更新日期:2020-07-20
  • Simple stably projectionless C*-algebras with generalized tracial rank one
    J. Noncommut. Geom. (IF 0.727) Pub Date : 2020-05-28
    George A. Elliott; Guihua Gong; Huaxin Lin; Zhuang Niu

    We study a class of stably projectionless simple C*-algebras which may be viewed as having generalized tracial rank one in analogy with the unital case. A number of structural questions concerning these simple C*-algebras are studied, pertinent to the classification of separable stably projectionless simple amenable Jiang–Su stable C*-algebras.

    更新日期:2020-07-20
  • Frobenius degenerations of preprojective algebras
    J. Noncommut. Geom. (IF 0.727) Pub Date : 2020-05-27
    Daniel Kaplan

    In this paper, we study a preprojective algebra for quivers decorated with $k$-algebras and bimodules, which generalizes work of Gabriel for ordinary quivers, work of Dlab and Ringel for$ $k-species, and recent work of de Thanhoffer de Völcsey and Presotto, which has recently appeared from a different perspective in work of Külshammer. As for undecorated quivers, we show that its moduli space of representations

    更新日期:2020-07-20
  • Quantum groups with projection and extensions of locally compact quantum groups
    J. Noncommut. Geom. (IF 0.727) Pub Date : 2020-04-30
    Paweł Kasprzak; Piotr M. Sołtan

    The main result of the paper is the characterization of those locally compact quantum groups with projection, i.e. non-commutative analogs of semidirect products, which are extensions as defined by L. Vainerman and S. Vaes. It turns out that quantum groups with projection are usually not extensions.We discuss several examples including the quantum $\mathrm{U}_q(2)$. The major tool used to obtain these

    更新日期:2020-04-30
  • The spectral localizer for even index pairings
    J. Noncommut. Geom. (IF 0.727) Pub Date : 2020-02-11
    Terry A. Loring; Hermann Schulz-Baldes

    Even index pairings are integer-valued homotopy invariants combining an even Fredholm module with a $K_0$-class specified by a projection. Numerous classical examples are known from differential and non-commutative geometry and physics. Here it is shown how to construct a finite-dimensional self-adjoint and invertible matrix, called the spectral localizer, such that half its signature is equal to the

    更新日期:2020-02-11
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