- journal articles metadata
European Mathematical Society Publishing House
2024-03-28 13:15:11
4
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=JNCG&vol=4&iss=4&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
4
2010
4
Double constructions of Frobenius algebras, Connes cocycles and their duality
Chengming
Bai
Nankai University, TIANJIN, CHINA
Associative algebra, Frobenius algebra, Connes cocycle, Yang–Baxter equation
We construct an associative algebra with a decomposition into the direct sum of the underlying vector spaces of another associative algebra and its dual space such that both of them are subalgebras and the natural symmetric bilinear form is invariant or the natural antisymmetric bilinear form is a Connes cocycle. The former is called a double construction of a Frobenius algebra and the latter is called a double construction of the Connes cocycle, which is interpreted in terms of dendriform algebras. Both of them are equivalent to a kind of bialgebras, namely, antisymmetric infinitesimal bialgebras and dendriform D-bialgebras, respectively. In the coboundary cases, our study leads to what we call associative Yang–Baxter equation in an associative algebra and D-equation in a dendriform algebra, respectively, which are analogues of the classical Yang–Baxter equation in a Lie algebra. We show that an antisymmetric solution of the associative Yang–Baxter equation corresponds to the antisymmetric part of a certain operator called O-operator which gives a double construction of a Frobenius algebra, whereas a symmetric solution of the D-equation corresponds to the symmetric part of an O-operator which gives a double construction of the Connes cocycle. By comparing antisymmetric infinitesimal bialgebras and dendriform D-bialgebras, we observe that there is a clear analogy between them. Due to the correspondences between certain symmetries and antisymmetries appearing in this analogy, we regard it as a kind of duality.
Associative rings and algebras
Nonassociative rings and algebras
Manifolds and cell complexes
Quantum theory
475
530
10.4171/JNCG/64
http://www.ems-ph.org/doi/10.4171/JNCG/64
Codes and noncommutative stochastic matrices
Sylvain
Lavallée
UQAM, MONTRÉAL, QC, CANADA
Dominique
Perrin
Université de Marne-la-Vallée, MARNE LA VALLÉE CEDEX 2, FRANCE
Vladimir
Retakh
Rutgers University, PISCATAWAY, UNITED STATES
Christophe
Reutenauer
UQAM, MONTRÉAL, QC, CANADA
Stochastic matrices, codes, eigenvectors, quasi-determinants
Given a matrix over a skew field fixing the column t(1,…,1), we give formulas for a row vector fixed by this matrix. The same techniques are applied to give noncommutative extensions of probabilistic properties of codes.
Associative rings and algebras
Combinatorics
General
531
554
10.4171/JNCG/65
http://www.ems-ph.org/doi/10.4171/JNCG/65
Quantum field theory on the degenerate Moyal space
Harald
Grosse
Universität Wien, WIEN, AUSTRIA
Fabien
Vignes-Tourneret
Université Claude Bernard Lyon 1, VILLEURBANNE CEDEX, FRANCE
Non-commutative geometry, quantum field theory, renormalization, Moyal space
We prove that the self-interacting scalar field on the four-dimensional degenerate Moyal plane is renormalisable to all orders when adding a suitable counterterm to the Lagrangian. Despite the apparent simplicity of the model, it raises several non-trivial questions. Our result is a first step towards the definition of renormalisable quantum field theories on a non-commutative Minkowski space.
Quantum theory
General
555
576
10.4171/JNCG/66
http://www.ems-ph.org/doi/10.4171/JNCG/66
The Heisenberg–Lorentz quantum group
Paweł
Kasprzak
Uniwersytet Warszawski, WARSAW, POLAND
Quantum groups, C*-algebras
In this article we present a new C*-algebraic deformation of the Lorentz group. It is obtained by means of the Rieffel deformation applied to SL(2,ℂ). We give a detailed description of the resulting quantum group $\mathbb{G}$ = (A,Δ) in terms of generators α, β, γ, δ ∈ Aη – the quantum counterparts of the matrix coefficients α, β, γ, δ of the fundamental representation of SL(2,ℂ). In order to construct β – the most involved of the four generators – we first define it on the quantum Borel subgroup $\mathbb G_0\subset\mathbb G$, then on the quantum complement of the Borel subgroup and finally we perform the gluing procedure. In order to classify representations of the C*-algebra A and to analyze the action of the comultiplication Δ on the generators α, β, γ, δ we employ the duality in the theory of locally compact quantum groups.
Functional analysis
Global analysis, analysis on manifolds
Topological groups, Lie groups
General
577
611
10.4171/JNCG/67
http://www.ems-ph.org/doi/10.4171/JNCG/67