- journal articles metadata
European Mathematical Society Publishing House
2024-03-28 14:04:50
11
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=JNCG&vol=12&iss=1&update_since=2024-03-28
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
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European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
12
2018
1
Index theory for manifolds with Baas–Sullivan singularities
Robin
Deeley
University of Colorado, Boulder, USA
$K$-homology, manifolds with Baas–Sullivan singularities, the Freed–Melrose index theorem
We study index theory for manifolds with Baas–Sullivan singularities using geometric $K$-homology with coefficients in a unital $C^*$-algebra. In particular, we define a natural analog of the Baum–Connes assembly map for a torsion-free discrete group in the context of these singular spaces. The cases of singularities modelled on $k$-points (i.e., $\mathbb Z/k\mathbb Z-manifolds) and the circle are discussed in detail. In the case of the former, the associated index theorem is related to the Freed–Melrose index theorem; in the case of the latter, the index theorem is related to work of Rosenberg.
$K$-theory
Functional analysis
Algebraic topology
1
28
10.4171/JNCG/269
http://www.ems-ph.org/doi/10.4171/JNCG/269
3
23
2018
Wreath products of finite groups by quantum groups
Amaury
Freslon
Université Paris-Sud, Orsay, France
Adam
Skalski
Polish Academy of Sciences, Warsaw, Poland
Quantum groups, wreath product, representation theory
We introduce a notion of partition wreath product of a finite group by a partition quantum group, a construction motivated on the one hand by classical wreath products and on the other hand by the free wreath product of J. Bichon.We identify the resulting quantum group in several cases, establish some of its properties and show that when the finite group in question is abelian, the partition wreath product is itself a partition quantum group. This allows us to compute its representation theory, using earlier results of the first named author.
Group theory and generalizations
Combinatorics
29
68
10.4171/JNCG/270
http://www.ems-ph.org/doi/10.4171/JNCG/270
3
23
2018
Hopf-dihedral (co)homology and $L$-theory
Atabey
Kaygun
Istanbul Technical University, Turkey
Serkan
Sütlü
Işık University, Istanbul, Turkey
Hopf $\ast$-algebras, $L$-theory, Chern character, Hopf-dihedral cohomology
We develop an appropriate dihedral extension of the Connes–Moscovici characteristic map for Hopf $\ast$-algebras. We then observe that one can use this extension together with the dihedral Chern character to detect non-trivial $L$-theory classes of a $\ast$-algebra that carry a Hopf symmetry over a Hopf $\ast$-algebra. Using our machinery we detect a previously unknown $L$-class of the standard Podleś sphere.
Associative rings and algebras
Category theory; homological algebra
69
106
10.4171/JNCG/271
http://www.ems-ph.org/doi/10.4171/JNCG/271
3
23
2018
Homotopic Hopf–Galois extensions revisited
Alexander
Berglund
Stockholm University, Sweden
Kathryn
Hess
Ecole Polytechnique Fédérale de Lausanne, Switzerland
Hopf–Galois extension, descent, Morita theory, model category
In this article we revisit the theory of homotopic Hopf–Galois extensions introduced in [9], in light of the homotopical Morita theory of comodules established in [3].We generalize the theory to a relative framework, which we believe is new even in the classical context and which is essential for treating the Hopf–Galois correspondence in [19]. We study in detail homotopic Hopf–Galois extensions of differential graded algebras over a commutative ring, for which we establish a descent-type characterization analogous to the one Rognes provided in the context of ring spectra [26]. An interesting feature in the differential graded setting is the close relationship between homotopic Hopf–Galois theory and Koszul duality theory. We show that nice enough principal fibrations of simplicial sets give rise to homotopic Hopf–Galois extensions in the differential graded setting, for which this Koszul duality has a familiar form.
Associative rings and algebras
Commutative rings and algebras
Category theory; homological algebra
Algebraic topology
107
155
10.4171/JNCG/272
http://www.ems-ph.org/doi/10.4171/JNCG/272
3
23
2018
An equivariant index for proper actions II: Properties and applications
Peter
Hochs
University of Adelaide, Australia
Yanli
Song
Washington University, St. Louis, USA
Equivariant index, proper group action, analytic $K$-homology
In the first part of this series, we defined an equivariant index without assuming the group acting or the orbit space of the action to be compact. This allowed us to generalise an index of deformed Dirac operators, defined for compact groups by Braverman. In this paper, we investigate properties and applications of this index. We prove that it has an induction property that can be used to deduce various other properties of the index. In the case of compact orbit spaces, the index is a special case of Kasparov’s index of transversally elliptic operators. In that case, we show how it is related to the analytic assembly map from the Baum–Connes conjecture, and an index used by Mathai and Zhang. In the case of noncompact orbit spaces, we use the index to define a notion of $K$-homological Dirac induction, and show that, under conditions, it satisfies the quantisation commutes with reduction principle.
Global analysis, analysis on manifolds
$K$-theory
157
193
10.4171/JNCG/273
http://www.ems-ph.org/doi/10.4171/JNCG/273
3
23
2018
The $C^*$-algebras of quantum lens and weighted projective spaces
Tomasz
Brzeziński
Swansea University, UK and University of Białystok, Poland
Wojciech
Szymański
University of Southern Denmark, Odense, Denmark
Quantum lens space, quantum weighted projective space, graph $C^*$-algebra
It is shown that the algebra of continuous functions on the quantum $2n+1$-dimensional lens space $C(L^{2n+1}_q(N; m_0,\ldots, m_n))$ is a graph $C^*$-algebra, for arbitrary positive weights $m_0,\ldots, m_n$. The form of the corresponding graph is determined from the skew product of the graph which defines the algebra of continuous functions on the quantum sphere $S_q^{2n+1}$ and the cyclic group $\mathbb Z_N$, with the labelling induced by the weights. Based on this description, the $K$-groups of specific examples are computed. Furthermore, the $K$-groups of the algebras of continuous functions on quantum weighted projective spaces $C(\mathbb W\mathbb P_q^n(m_0,\ldots, m_n))$, interpreted as fixed points under the circle action on $C(S_q^{2n+1})$, are computed under a mild assumption on the weights.
Functional analysis
Global analysis, analysis on manifolds
195
215
10.4171/JNCG/274
http://www.ems-ph.org/doi/10.4171/JNCG/274
3
23
2018
Automorphisms of Cuntz–Krieger algebras
Søren
Eilers
University of Copenhagen, Denmark
Gunnar
Restorff
University of the Faroe Islands, Tórshavn, Faroe Islands
Efren
Ruiz
University of Hawaii, Hilo, USA
KK-theory, UCT, Cuntz–Krieger algebras, automorphisms
We prove that the natural homomorphism from Kirchberg’s ideal-related $KK$-theory, $KK_\mathcal E(e, e')$, with one specified ideal, into $\mathrm{Hom}_{\Lambda} (\ushort{K}_{\mathcal{E}} (e), \ushort{K}_{\mathcal{E}} (e'))$ is an isomorphism for all extensions $e$ and $e'$ of separable, nuclear $C^{*}$-algebras in the bootstrap category $\mathcal{N}$ with the $K$-groups of the associated cyclic six term exact sequence being finitely generated, having zero exponential map and with the $K_{1}$-groups of the quotients being free abelian groups. This class includes all Cuntz–Krieger algebras with exactly one non-trivial ideal. Combining our results with the results of Kirchberg, we classify automorphisms of the stabilized purely infinite Cuntz–Krieger algebras with exactly one non-trivial ideal modulo asymptotically unitary equivalence. We also get a classification result modulo approximately unitary equivalence. The results in this paper also apply to certain graph algebras.
Functional analysis
217
254
10.4171/JNCG/275
http://www.ems-ph.org/doi/10.4171/JNCG/275
3
23
2018
Two problems from the Polishchuk and Positselski book on quadratic algebras
Natalia
Iyudu
The University of Edinburgh, UK
Stanislav
Shkarin
Queen's University Belfast, UK
Quadratic algebras, Koszul algebras, Hilbert series, Gröbner basis
In the book Quadratic algebras by Polishchuk and Positselski [23], algebras with a small number of generators $(n=2,3)$ are considered. For some number $r$ of relations possible Hilbert series are listed, and those appearing as series of Koszul algebras are specified. The first case, where it was not possible to do, namely the case of three generators $n=3$ and six relations $r=6$ is formulated as an open problem. We give here a complete answer to this question, namely for quadratic algebras with $\mathrm {dim} A_1=\mathrm {dim} A_2=3$, we list all possible Hilbert series, and find out which of them can come from Koszul algebras, and which can not. As a consequence of this classification, we found an algebra, which serves as a counterexample to another problem from the same book [23, Chapter 7, Sec. 1, Conjecture 2], saying that Koszul algebra of finite global homological dimension $d$ has $\mathrm {dim} A_1 \geq d$. Namely, the 3-generated algebra $A$ given by relations $xx+yx=xz=zy=0$ is Koszul and its Koszul dual algebra $A^!$ has Hilbert series of degree 4: $H_{A^!}(t)= 1+3t+3t^2+2t^3+t^4$, hence $A$ has global homological dimension 4.
Associative rings and algebras
Algebraic geometry
255
278
10.4171/JNCG/276
http://www.ems-ph.org/doi/10.4171/JNCG/276
3
23
2018
The maximal quantum group-twisted tensor product of C*-algebras
Sutanu
Roy
National Institute of Science Education and Research (NISER), Jatni, India
Thomas
Timmermann
Universität Münster, Germany
C*-algebra, tensor product, crossed product, quantum group
We construct a maximal counterpart to the minimal quantum group-twisted tensor product of C *-algebras studied by Meyer, Roy and Woronowicz [16, 17], which is universal with respect to representations satisfying certain braided commutation relations. Much like the minimal one, this product yields a monoidal structure on the coactions of a quasi-triangular C *-quantum group, the horizontal composition in a bicategory of Yetter–Drinfeld C *-algebras, and coincides with a Rieffel deformation of the non-twisted tensor product in the case of group coactions.
Quantum theory
Functional analysis
279
330
10.4171/JNCG/277
http://www.ems-ph.org/doi/10.4171/JNCG/277
3
23
2018
Graded twisting of comodule algebras and module categories
Julien
Bichon
Université Clermont Auvergne, Aubière, France
Sergey
Neshveyev
University of Oslo, Norway
Makoto
Yamashita
Ochanomizu University, Tokyo, Japan
Hopf algebras, comodule algebras, coinvariants, monoidal categories, module categories over monoidal categories, Poisson boundary
Continuing our previous work on graded twisting of Hopf algebras and monoidal categories, we introduce a graded twisting construction for equivariant comodule algebras and module categories. As an example we study actions of quantum subgroups of $G\subset\mathrm {SL}_{-1}(2)$ on $K_{-1}[x,y]$ and show that in most cases the corresponding invariant rings $K_{-1}[x,y]^G$ are invariant rings $K[x,y]^{G'}$ for the action of a classical subgroup $G'\subset \mathrm {SL}(2)$. As another example we study Poisson boundaries of graded twisted categories and show that under the assumption of weak amenability they are graded twistings of the Poisson boundaries.
Associative rings and algebras
Category theory; homological algebra
331
368
10.4171/JNCG/278
http://www.ems-ph.org/doi/10.4171/JNCG/278
3
23
2018
Strongly self-absorbing C*-dynamical systems. II
Gábor
Szabó
Universität Münster, Germany and Copenhagen University, Denmark
Noncommutative dynamics, strongly self-absorbing C*-algebra
This is a continuation of the study of strongly self-absorbing actions of locally compact groups on C -algebras. Given a strongly self-absorbing action $\gamma: G \curvearrowright \mathcal{D}$, we establish permanence properties for the class of separable C* -dynamical systems absorbing $\gamma$ tensorially up to cocycle conjugacy. Generalizing results of both Toms–Winter and Dadarlat–Winter, it is proved that the desirable equivariant analogues of the classical permanence properties hold in this context. For the permanence with regard to equivariant extensions, we need to require a mild extra condition on $\gamma$, which replaces $K_1$-injectivity assumptions in the classical theory. This condition turns out to be automatic for equivariantly Jiang–Su absorbing C* -dynamical systems, yielding a large class of examples. It is left open whether this condition is redundant for all strongly self-absorbing actions, and we do consider examples that satisfy this condition but are not equivariantly Jiang–Su absorbing.
Functional analysis
369
406
10.4171/JNCG/279
http://www.ems-ph.org/doi/10.4171/JNCG/279
3
23
2018