scholarly journals Cyclic cohomology and Baaj-Skandalis duality

2013 ◽  
Vol 13 (1) ◽  
pp. 115-145 ◽  
Author(s):  
Christian Voigt
Keyword(s):  
Quantum Group ◽  
Lie Groups ◽  
Quantum Groups ◽  
Hopf Algebras ◽  
Cyclic Homology ◽  
Cyclic Cohomology ◽  
Important Ingredient ◽  
Kasparov Theory

AbstractWe construct a duality isomorphism in equivariant periodic cyclic homology analogous to Baaj-Skandalis duality in equivariant Kasparov theory. As a consequence we obtain general versions of the Green-Julg theorem and the dual Green-Julg theorem in periodic cyclic theory.Throughout we work within the framework of bornological quantum groups, thus in particular incorporating at the same time actions of arbitrary classical Lie groups as well as actions of compact or discrete quantum groups. An important ingredient in the construction of our duality isomorphism is the notion of a modular pair for a bornological quantum group, closely related to the concept introduced by Connes and Moscovici in their work on cyclic cohomology for Hopf algebras.

scholarly journals INTRODUCTION TO QUANTIZED LIE GROUPS AND ALGEBRAS

1992 ◽  
Vol 07 (25) ◽  
pp. 6175-6213 ◽  
Author(s):  
T. TJIN
Keyword(s):  
Lie Groups ◽  
Quantum Groups ◽  
Hopf Algebras ◽  
Baxter Equation ◽  
Product Decomposition ◽  
R Matrix ◽  
Conformal Field ◽  
Finite Dimensional ◽  
Lie Groups And Algebras ◽  
Poisson Lie Groups

We give a self-contained introduction to the theory of quantum groups according to Drinfeld, highlighting the formal aspects as well as the applications to the Yang-Baxter equation and representation theory. Introductions to Hopf algebras, Poisson structures and deformation quantization are also provided. After defining Poisson Lie groups we study their relation to Lie bialgebras and the classical Yang-Baxter equation. Then we explain in detail the concept of quantization for them. As an example the quantization of sl2 is explicitly carried out. Next we show how quantum groups are related to the Yang-Baxter equation and how they can be used to solve it. Using the quantum double construction we explicitly construct the universal R matrix for the quantum sl2 algebra. In the last section we deduce all finite-dimensional irreducible representations for q a root of unity. We also give their tensor product decomposition (fusion rules), which is relevant to conformal field theory.

The dual of a certain left quantum group

2016 ◽  
Vol 15 (04) ◽  
pp. 1650072 ◽  
Author(s):  
Uma N. Iyer ◽  
Earl J. Taft
Keyword(s):  
Hopf Algebra ◽  
Quantum Group ◽  
Quantum Groups ◽  
Hopf Algebras ◽  
Current Work ◽  
Independent Interest ◽  
Natural Choice

The second author and his collaborators, Lauve and Rodriguez-Romo (see [A class of left quantum groups modeled after SL[Formula: see text]([Formula: see text]), J. Pure Appl. Algebra 208(3) (2007) 797–803; A left quantum group, J. Algebra 286 (2005) 154–160), have constructed left Hopf algebras which are not Hopf algebras modeled after [Formula: see text]. In particular, they constructed the left quantum group [Formula: see text], along with an epimorphism to the quantum group [Formula: see text], the latter being a Hopf algebra. The current work began as a search for a left Hopf algebra (which is not a Hopf algebra) containing the quantum group [Formula: see text], the latter being a Hopf algebra. The natural choice was to look for the dual of [Formula: see text]. We show that the Hopf dual of [Formula: see text] is equal to the Hopf dual of [Formula: see text], which is of independent interest. Thus the Hopf dual of [Formula: see text] is a Hopf algebra. The original search of a left Hopf algebra which is not a Hopf algebra, larger than [Formula: see text], is still open.

scholarly journals A CLOSED EXPRESSION FOR THE UNIVERSAL ℛ-MATRIX IN A NON-STANDARD QUANTUM DOUBLE

1993 ◽  
Vol 08 (27) ◽  
pp. 2573-2578 ◽  
Author(s):  
A.A. VLADIMIROV
Keyword(s):  
Quantum Group ◽  
Lie Algebra ◽  
Quantum Groups ◽  
Hopf Algebras ◽  
Reduced Form ◽  
Elementary Functions ◽  
Quantum Double ◽  
Closed Expression ◽  
Standard Quantum ◽  
Number Of Generators

In recent papers by the author, a method was developed for constructing quasitriangular Hopf algebras (quantum groups) of the quantum-double type. As a by-product, a novel non-standard example of the quantum double has been found. In this letter, a closed expression (in terms of elementary functions) for the corresponding universal ℛ-matrix is obtained. In reduced form, when the number of generators becomes two instead of four, this quantum group can be interpreted as a deformation of the Lie algebra [x, h]=2h in the context of Drinfeld’s quantization program.

Compact Hopf *-Algebras, Quantum Enveloping Algebras and Dual Woronowicz Algebras for Quantum Lorentz Groups

1997 ◽  
Vol 08 (07) ◽  
pp. 959-997 ◽  
Author(s):  
Hideki Kurose ◽  
Yoshiomi Nakagami
Keyword(s):  
Hopf Algebra ◽  
Quantum Group ◽  
Quantum Groups ◽  
Hopf Algebras ◽  
Canonical Representation ◽  
Compact Quantum Group ◽  
Enveloping Algebras ◽  
Enveloping Algebra ◽  
Quantum Enveloping Algebra ◽  
Definition Of

A compact Hopf *-algebra is a compact quantum group in the sense of Koornwinder. There exists an injective functor from the category of compact Hopf *-algebras to the category of compact Woronowicz algebras. A definition of the quantum enveloping algebra Uq(sl(n,C)) is given. For quantum groups SUq(n) and SLq(n,C), the commutant of a canonical representation of the quantum enveloping algebra for q coincides with the commutant of the dual Woronowicz algebra for q-1.

scholarly journals RIESZ TRANSFORMS ON COMPACT QUANTUM GROUPS AND STRONG SOLIDITY

2021 ◽  
pp. 1-37
Author(s):  
Martijn Caspers
Keyword(s):  
Quantum Group ◽  
Quantum Groups ◽  
Riesz Transform ◽  
Wreath Products ◽  
Compact Quantum Group ◽  
Riesz Transforms ◽  
Markov Semigroup ◽  
Markov Semigroups ◽  
Free Products ◽  
Compact Quantum Groups

Abstract One of the main aims of this paper is to give a large class of strongly solid compact quantum groups. We do this by using quantum Markov semigroups and noncommutative Riesz transforms. We introduce a property for quantum Markov semigroups of central multipliers on a compact quantum group which we shall call ‘approximate linearity with almost commuting intertwiners’. We show that this property is stable under free products, monoidal equivalence, free wreath products and dual quantum subgroups. Examples include in particular all the (higher-dimensional) free orthogonal easy quantum groups. We then show that a compact quantum group with a quantum Markov semigroup that is approximately linear with almost commuting intertwiners satisfies the immediately gradient- ${\mathcal {S}}_2$ condition from [10] and derive strong solidity results (following [10]). Using the noncommutative Riesz transform we also show that these quantum groups have the Akemann–Ostrand property; in particular, the same strong solidity results follow again (now following [27]).

On codihedral module for *-Hopf algebras

2008 ◽  
Vol 2 (2) ◽  
pp. 353-360
Author(s):  
Th. Yu. Popelensky
Keyword(s):  
Hopf Algebras ◽  
Cyclic Cohomology

AbstractWe construct dihedral and reflexive cohomology theories for *-Hopf algebras. This generalizes the Connes–Moscovici construction of cyclic cohomology for Hopf algebras.

POISSON–LIE GROUPS AND CLASSICAL W-ALGEBRAS

1992 ◽  
Vol 07 (05) ◽  
pp. 853-876 ◽  
Author(s):  
V. A. FATEEV ◽  
S. L. LUKYANOV
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Quantum Group ◽  
Lie Groups ◽  
Group Structure ◽  
Two Dimensional ◽  
Poisson Structures ◽  
Additional Symmetries ◽  
Conformal Field ◽  
Poisson Lie Groups

This is the first part of a paper studying the quantum group structure of two-dimensional conformal field theory with additional symmetries. We discuss the properties of the Poisson structures possessing classical W-invariance. The Darboux variables for these Poisson structures are constructed.

scholarly journals From Quantum Groups to Groups

2013 ◽  
Vol 65 (5) ◽  
pp. 1073-1094 ◽  
Author(s):  
Mehrdad Kalantar ◽  
Matthias Neufang
Keyword(s):  
Quantum Group ◽  
Quantum Groups ◽  
Locally Compact ◽  
Large Classes ◽  
Locally Compact Quantum Groups ◽  
Locally Compact Quantum Group ◽  
Recent Developments ◽  
Compact Quantum Groups ◽  
Quantum Version ◽  
Quantum Point

AbstractIn this paper we use the recent developments in the representation theory of locally compact quantum groups, to assign to each locally compact quantum group 𝔾 a locally compact group 𝔾˜ that is the quantum version of point-masses and is an invariant for the latter. We show that “quantum point-masses” can be identified with several other locally compact groups that can be naturally assigned to the quantum group 𝔾. This assignment preserves compactness as well as discreteness (hence also finiteness), and for large classes of quantum groups, amenability. We calculate this invariant for some of the most well-known examples of non-classical quantum groups. Also, we show that several structural properties of 𝔾 are encoded by 𝔾˜; the latter, despite being a simpler object, can carry very important information about 𝔾.

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