- journal articles metadata
European Mathematical Society Publishing House
2024-03-29 10:49:04
10
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=JNCG&vol=7&iss=3&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
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European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
7
2013
3
Star product realizations of κ-Minkowski space
Bergfinnur
Durhuus
University of Copenhagen, COPENHAGEN, DENMARK
Andrzej
Sitarz
Jagiellonian University, KRAKOW, POLAND
Quantization, deformation, star product, kappa-Minkowski space, quantum Poincaré algebra
We define a family of star products and involutions associated with $\kappa$-Minkowski space. Applying corresponding quantization maps we show that these star products restricted to a certain space of Schwartz functions have isomorphic Banach algebra completions. For two particular star products it is demonstrated that they can be extended to a class of polynomially bounded smooth functions allowing a realization of the full Hopf algebra structure on $\kappa$-Minkowski space. Furthermore, we give an explicit realization of the action of the $\kappa$-Poincaré algebra as an involutive Hopf algebra on this representation of $\kappa$-Minkowski space and initiate a study of its properties.
Functional analysis
Differential geometry
Quantum theory
605
645
10.4171/JNCG/129
http://www.ems-ph.org/doi/10.4171/JNCG/129
The odd-dimensional analogue of a theorem of Getzler and Wu
Zhizhang
Xie
Vanderbilt University, NASHVILLE, UNITED STATES
Odd dimensional manifolds with boundary, b-calculus, noncommutative geometry, Connes–Chern character, odd APS index
We prove an analogue for odd-dimensional manifolds with boundary, in the b-calculus setting, of the higher Atiyah–Patodi–Singer index theorem by Getzler and by Wu, and thus obtain a natural counterpart of the eta invariant for even-dimensional closed manifolds.
Global analysis, analysis on manifolds
Functional analysis
647
676
10.4171/JNCG/130
http://www.ems-ph.org/doi/10.4171/JNCG/130
Nonperturbative spectral action of round coset spaces of SU(2)
Kevin
Teh
California Institute of Technology, PASADENA, UNITED STATES
Noncommutative geometry, spectral action principle, Dirac spectrum, spectral geometry, lens space, Poincaré homology sphere, Berger metric, 3-sphere
We compute the spectral action of SU(2)/$\Gamma$ with the trivial spin structure and the round metric and find it in each case to be equal to $\frac{1}{\vert \Gamma \vert}(\Lambda^3 \hat{f}^{(2)}(0) - \frac{1}{4}\Lambda \hat{f}(0) )+ O(\Lambda^{-\infty})$. We do this by explicitly computing the spectrum of the Dirac operator for SU(2)/$\Gamma$ equipped with the trivial spin structure and a selection of metrics. Here $\Gamma$ is a finite subgroup of SU(2). In the case where $\Gamma$ is cyclic, or dicyclic, we consider the one-parameter family of Berger metrics, which includes the round metric, and when $\Gamma$ is the binary tetrahedral, binary octahedral or binary icosahedral group, we only consider the case of the round metric.
Global analysis, analysis on manifolds
General
677
708
10.4171/JNCG/131
http://www.ems-ph.org/doi/10.4171/JNCG/131
An equivariant noncommutative residue
Shantanu
Dave
Universität Wien, WIEN, AUSTRIA
Noncommutative residue, cyclic homology, cross-product algebra
Let $\Gamma $ be a finite group acting on a compact manifold $M$ and let $\mathcal{A}(M)$ denote the algebra of classical complete symbols on $M$. We determine all traces on the cross-product algebra $\mathcal{A}(M) \rtimes \Gamma$ as residues of certain meromorphic zeta functions. Further we compute the cyclic homology for $\mathcal{A}(M)\rtimes\Gamma $ in terms of the de Rham cohomology of the fixed point manifolds $S^*M^g$. In the process certain new results on the homologies of general cross-product algebras are obtained.
Global analysis, analysis on manifolds
General
709
735
10.4171/JNCG/132
http://www.ems-ph.org/doi/10.4171/JNCG/132
Path subcoalgebras, finiteness properties and quantum groups
Sorin
Dăscălescu
University of Bucharest, BUCHAREST, ROMANIA
Miodrag
Iovanov
University of Southern California, LOS ANGELES, UNITED STATES
Constantin
Năstăsescu
University of Bucharest, BUCHAREST, ROMANIA
Incidence coalgebra, path coalgebra, co-Frobenius coalgebra, quasi-co-Frobenius coalgebra, balanced bilinear form, quantum group, integral
We study subcoalgebras of path coalgebras that are spanned by paths (called path subcoalgebras) and subcoalgebras of incidence coalgebras, and propose a unifying approach for these classes. We discuss the left quasi-co-Frobenius and the left co-Frobenius properties for these coalgebras. We classify the left co-Frobenius path subcoalgebras, showing that they are direct sums of certain path subcoalgebras arising from the infinite line quiver or from cyclic quivers. We investigate which of the co-Frobenius path subcoalgebras can be endowed with Hopf algebra structures, in order to produce some quantum groups with non-zero integrals, and we classify all these structures over a field with primitive roots of unity of any order. These turn out to be liftings of quantum lines over certain not necessarily abelian groups.
Associative rings and algebras
Order, lattices, ordered algebraic structures
737
766
10.4171/JNCG/133
http://www.ems-ph.org/doi/10.4171/JNCG/133
Chow motives versus noncommutative motives
Gonçalo
Tabuada
Massachusetts Institute of Technology, CAMBRIDGE, UNITED STATES
Chow motives, noncommutative motives, Kimura and Schur finiteness, motivic measures, motivic zeta functions
In this article we formalize and enhance Kontsevich’s beautiful insight that Chow motives can be embedded into noncommutative ones after factoring out by the action of the Tate object. We illustrate the potential of this result by developing three of its manyfold applications: (1) the notions of Schur and Kimura finiteness admit an adequate extension to the realm of noncommutative motives; (2) Gillet–Soulé’s motivic measure admits an extension to the Grothendieck ring of noncommutative motives; (3) certain motivic zeta functions admit an intrinsic construction inside the category of noncommutative motives.
Algebraic geometry
General
$K$-theory
767
786
10.4171/JNCG/134
http://www.ems-ph.org/doi/10.4171/JNCG/134
Classification of spin structures on the noncommutative n-torus
Jan Jitse
Venselaar
Caltech, PASDENA, UNITED STATES
Spectral geometry, noncommutative geometry, real spectral triples
We classify spin structures on the noncommutative torus and find that the noncommutative $n$-torus has $2^n$ spin structures corresponding to isospectral deformations of spin structures on the commutative $n$-torus. For $n\geq 4$ the classification depends on Connes’ spin manifold theorem. In addition, we study unitary equivalences of these spin structures.
Global analysis, analysis on manifolds
Functional analysis
787
816
10.4171/JNCG/135
http://www.ems-ph.org/doi/10.4171/JNCG/135
Noncommutative quadric surfaces
S. Paul
Smith
University of Washington, SEATTLE, UNITED STATES
Michel
Van den Bergh
Hasselt University, HASSELT, BELGIUM
Noncommutative algebraic geometry, noncommutative quadric surfaces, Sklyanin algebra
The 4-dimensional Sklyanin algebra is the homogeneous coordinate ring of a noncommutative analogue of projective 3-space. The degree-two component of the algebra contains a 2-dimensional subspace of central elements. The zero loci of those central elements, except 0, form a pencil of noncommutative quadric surfaces. We show that the behavior of this pencil is similar to that of a generic pencil of quadrics in the commutative projective 3-space. There are exactly four singular quadrics in the pencil. The singular and non-singular quadrics are characterized by whether they have one or two rulings by noncommutative lines. The Picard groups of the smooth quadrics are free abelian of rank two. The alternating sum of dimensions of Ext groups allows us to define an intersection pairing on the Picard group of the smooth noncommutative quadrics. A surprise is that a smooth noncommutative quadric can sometimes contain a “curve” having self-intersection number −2. Many of the methods used in our paper are noncommutative versions of methods developed by Buchweitz, Eisenbud and Herzog: in particular, the correspondence between the geometry of a quadric hypersurface and maximal Cohen–Macaulay modules over its homogeneous coordinate ring plays a key role. An important aspect of our work is to introduce definitions of noncommutative analogues of the familiar commutative terms used in this abstract. We expect the ideas we develop here for 2-dimensional noncommutative quadric hypersurfaces will apply to higher dimensional noncommutative quadric hypersurfaces and we develop them in sufficient generality to make such applications possible.
Algebraic geometry
General
817
856
10.4171/JNCG/136
http://www.ems-ph.org/doi/10.4171/JNCG/136
The closed state space of affine Landau–Ginzburg B-models
Ed
Segal
Imperial College London, LONDON, UNITED KINGDOM
Landau–Ginzburg models, Hochschild homology, B-model, matrix factorizations
We study the category of perfect cdg-modules over a curved algebra, and in particular the category of B-branes in an affine Landau–Ginzburg model. We construct an explicit chain map from the Hochschild complex of the category to the closed state space of the model, and prove that this is a quasi-isomorphism from the Borel–Moore Hochschild complex. Using the lowest-order term of our map we derive Kapustin and Li’s formula for the correlator of an open-string state over a disc.
Global analysis, analysis on manifolds
General
857
883
10.4171/JNCG/137
http://www.ems-ph.org/doi/10.4171/JNCG/137
Hopf cyclic cohomology and Hodge theory for proper actions
Xiang
Tang
Washington University, ST. LOUIS, UNITED STATES
Yi-Jun
Yao
Fudan University, SHANGHAI, CHINA
Weiping
Zhang
Nankai University, TIANJIN, CHINA
Cyclic cohomology, Hodge theory, proper action, Euler characteristic
We introduce a Hopf algebroid associated to a proper Lie group action on a smooth manifold. We prove that the cyclic cohomology of this Hopf algebroid is equal to the de Rham cohomology of invariant differential forms. When the action is cocompact, we develop a generalized Hodge theory for the de Rham cohomology of invariant differential forms. We prove that every cyclic cohomology class of the Hopf algebroid is represented by a generalized harmonic form. This implies that the space of cyclic cohomology of the Hopf algebroid is finite dimensional. As an application of the techniques developed in this paper, we discuss properties of the Euler characteristic for a proper cocompact action.
Global analysis, analysis on manifolds
Differential geometry
885
905
10.4171/JNCG/138
http://www.ems-ph.org/doi/10.4171/JNCG/138