PlanetMath: locally compact quantum group

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locally compact quantum group

(Definition)

Definition 0.1 A locally compact quantum group defined as in ref. [1] is a quadruple $\mathcal G= (A, \Delta, \mu, \nu)$ , where

is either a C*- or a - algebra equipped with a co-associative comultiplication $\Delta: A \to A \otimes A$ and two faithful semi-finite normal weights, $\mu$ and $\nu$ -right and -left Haar measures.

Examples

An ordinary unimodular group with Haar measure $\mu$ . $A := L^{\infty}(G, \mu), \Delta: f(g) \mapsto f(gh)$ , $S: f(g) \mapsto f(g^{}-1), \phi(f) = \int_G f(g)d\mu (g)$ , where $g, h \in G, f \in L^{\infty}(G, \mu)$ .
A:= Ł(G) is the von Neumann algebra generated by left-translations or by left convolutions $L_f :={ \int}_G f(g)L_g d \mu (g)$ with continuous functions $f(.) \in L^1(G,\mu) \Delta: \mapsto L_g \otimes L_g \mapsto L_g^{-1}, \phi(f) = f(e)$ , where $g \in G$ , and e is the unit of G.

Bibliography

1: Leonid Vainerman. 2003. Locally Compact Quantum Groups and Groupoids: Proceedings of the Meeting of Theoretical Physicists and Mathematicians, Strasbourg, February 21-23, 2002., Series in Mathematics and Theoretical Physics, 2, Series ed. V. Turaev., Walter de Gruyter Gmbh & Co: Berlin.

"locally compact quantum group" is owned by bci1.

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Other names:

Hopf algebras, ring groups

Also defines:

quantum group, local quantum symmetry

Keywords:

quantum group, local quantum symmetry, C*-algebra, $W^*$

-algebra, Hopf algebras, ring groups, quadruple, coassociative multiplication, locally compact quantum groups, quantum groupoids, discrete quantum groups, algebraic quantum groups, monoidal category structure, unitary corepresentations, weak bialgebras, relative tensor product, multiplicative unitary, corresponding matched pair, faithful weight, multiplicative unitaries, quantum subgroup, involutive homomorphism, face algebra, grouplike elements, partial isometrics, partial isometry, infinitesimal objects, quantum group theory, hase change, monoidal functor, canonical trace, preprint math, dense ideal

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Cross-references: unit, continuous functions, convolutions, generated by, von Neumann algebra, unimodular group, Haar measures, right, weights, normal, faithful
There are 17 references to this entry.

This is version 9 of locally compact quantum group, born on 2008-09-06, modified 2008-09-21.
Object id is 10994, canonical name is LocallyCompactQuantumGroup.
Accessed 523 times total.

Classification:

AMS MSC:	22D25 (Topological groups, Lie groups :: Locally compact groups and their algebras :: $C^$-algebras and $W$-algebras in relation to group representations)
	46L05 (Functional analysis :: Selfadjoint operator algebras :: General theory of $C^*$-algebras)
	17B37 (Nonassociative rings and algebras :: Lie algebras and Lie superalgebras :: Quantum groups and related deformations)
	22A22 (Topological groups, Lie groups :: Topological and differentiable algebraic systems :: Topological groupoids )
	18B40 (Category theory; homological algebra :: Special categories :: Groupoids, semigroupoids, semigroups, groups )
	46M20 (Functional analysis :: Methods of category theory in functional analysis :: Methods of algebraic topology )
	81R50 (Quantum theory :: Groups and algebras in quantum theory :: Quantum groups and related algebraic methods)

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