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Transactions of the American Mathematical Society

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Complex quantum groups and a deformation of the Baum-Connes assembly map


Authors: Andrew Monk and Christian Voigt
Journal: Trans. Amer. Math. Soc. 371 (2019), 8849-8877
MSC (2010): Primary 20G42, 46L80; Secondary 46L65
DOI: https://doi.org/10.1090/tran/7774
Published electronically: February 11, 2019
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Abstract: We define and study an analogue of the Baum-Connes assembly map for complex semisimple quantum groups, that is, Drinfeld doubles of $ q $-deformations of compact semisimple Lie groups.

Our starting point is the deformation picture of the Baum-Connes assembly map for a complex semisimple Lie group $ G $, which allows one to express the $ K$-theory of the reduced group $ C^* $-algebra of $ G $ in terms of the $ K $-theory of its associated Cartan motion group. The latter can be identified with the semidirect product of the maximal compact subgroup $ K $ acting on $ \mathfrak{k}^* $ via the coadjoint action.

In the quantum case the role of the Cartan motion group is played by the Drinfeld double of the classical group $ K $, whose associated group $ C^* $-algebra is the crossed product of $ C(K) $ with respect to the adjoint action of $ K $. Our quantum assembly map is obtained by varying the deformation parameter in the Drinfeld double construction applied to the standard deformation $ K_q $ of $ K $. We prove that the quantum assembly map is an isomorphism, thus providing a description of the $ K $-theory of complex quantum groups in terms of classical topology.

Moreover, we show that there is a continuous field of $ C^* $-algebras which encodes both the quantum and classical assembly maps as well as a natural deformation between them. It follows in particular that the quantum assembly map contains the classical Baum-Connes assembly map as a direct summand.


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Additional Information

Andrew Monk
Affiliation: School of Mathematics and Statistics, University of Glasgow, University Place, Glasgow G12 8SQ, United Kingdom
Email: a.monk.1@research.gla.ac.uk

Christian Voigt
Affiliation: School of Mathematics and Statistics, University of Glasgow, University Place, Glasgow G12 8SQ, United Kingdom
Email: christian.voigt@glasgow.ac.uk

DOI: https://doi.org/10.1090/tran/7774
Keywords: Quantum groups, Baum--Connes conjecture
Received by editor(s): October 5, 2018
Received by editor(s) in revised form: November 28, 2018
Published electronically: February 11, 2019
Additional Notes: The second author was supported by the Polish National Science Centre grant no. 2012/06/M/ST1/00169.
This work was supported by EPSRC grant no. EP/K032208/1.
The authors would like to thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, England, for support and hospitality during the program Operator Algebras: Subfactors and Their Applications, where work on this paper was undertaken.
Article copyright: © Copyright 2019 American Mathematical Society