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# Categorical Aspects of Quantum Groups: Multipliers and Intrinsic Groups

Published:2016-01-22
Printed: Apr 2016
• Matthew Daws,
School of Mathematics, University of Leeds, Leeds LS2 9JT, UK
 Format: LaTeX MathJax PDF

## Abstract

We show that the assignment of the (left) completely bounded multiplier algebra $M_{cb}^l(L^1(\mathbb G))$ to a locally compact quantum group $\mathbb G$, and the assignment of the intrinsic group, form functors between appropriate categories. Morphisms of locally compact quantum groups can be described by Hopf $*$-homomorphisms between universal $C^*$-algebras, by bicharacters, or by special sorts of coactions. We show that the whole theory of completely bounded multipliers can be lifted to the universal $C^*$-algebra level, and that then the different pictures of both multipliers (reduced, universal, and as centralisers) and morphisms interact in extremely natural ways. The intrinsic group of a quantum group can be realised as a class of multipliers, and so our techniques immediately apply. We also show how to think of the intrinsic group using the universal $C^*$-algebra picture, and then, again, show how the differing views on the intrinsic group interact naturally with morphisms. We show that the intrinsic group is the maximal classical'' quantum subgroup of a locally compact quantum group, show that it is even closed in the strong Vaes sense, and that the intrinsic group functor is an adjoint to the inclusion functor from locally compact groups to quantum groups.
 Keywords: locally compact quantum group, morphism, intrinsic group, multiplier, centraliser
 MSC Classifications: 20G42 - Quantum groups (quantized function algebras) and their representations [See also 16T20, 17B37, 81R50] 22D25 - $C^*$-algebras and $W^*$-algebras in relation to group representations [See also 46Lxx] 43A22 - Homomorphisms and multipliers of function spaces on groups, semigroups, etc. 43A35 - Positive definite functions on groups, semigroups, etc. 43A95 - Categorical methods [See also 46Mxx] 46L52 - Noncommutative function spaces 46L89 - Other noncommutative'' mathematics based on $C^*$-algebra theory [See also 58B32, 58B34, 58J22] 47L25 - Operator spaces (= matricially normed spaces) [See also 46L07]

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