Complex quantum groups and a deformation of the Baum–Connes assembly map
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- by Andrew Monk and Christian Voigt PDF
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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
- MR Author ID: 819355
- Email: christian.voigt@glasgow.ac.uk
- 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. - © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 371 (2019), 8849-8877
- MSC (2010): Primary 20G42, 46L80; Secondary 46L65
- DOI: https://doi.org/10.1090/tran/7774
- MathSciNet review: 3955567