# Compact Quantum Groups

This is a reading list for the series of talks which I'm giving in Leeds in November. The links are to outside websites, but the links [Leeds] will take you via the Leeds Library which will allow anyone to view the website from any location, provided they have a Leeds University IT login, or Library card.

This is very far from a complete bibliography, but should give some good sources to get started with.

### Core Material

- S.L. Woronowicz, Compact Quantum Groups
[Leeds],
Symetries quantiques (Les Houches, 1995), 845--884, North-Holland, Amsterdam, 1998.

This is the original paper, but it is quite hard to get hold of. - A. Maes, A. Van Daele, Notes on Compact Quantum Groups
[Leeds],
Nieuw Arch. Wisk. (4) 16 (1998), no. 1-2, 73--112.

This is a nice survey paper, which is both easier to read than the above, and more easily available, as it is on the arXiv: arXiv:math/9803122v1. - S.L. Woronowicz, Compact matrix pseudogroups
[Leeds],
Comm. Math. Phys.
[Leeds]
111 (1987), no. 4, 613--665.

This is an older paper, where the axiom are not quite in final form. However, it is excellent background reading, and provides many important techniques. - G.J. Murphy, L. Tuset, Aspects of compact quantum group theory
[Leeds],
Proc. Amer. Math. Soc.
[Leeds]
132 (2004), no. 10, 3055--3067.

This is a nice paper showing the existance of the Haar State, and showing that weaker axioms than Woronowicz's will work.

### Further reading

- E.C. Lance, Hilbert $C\sp *$-modules
[Leeds],
London Mathematical Society Lecture Note Series, 210. Cambridge University Press, Cambridge, 1995.

A nice little book by our very own Chris Lance. If you get at all far into quantum group theory in the C*-setting, and don't want to use von Neumann algebras, then you will have to deal with multiplier algebras. Hilbert C*-modules give a good framework to study multipliers in. - J. Kustermans, S. Vaes, Locally compact quantum groups
[Leeds],
Ann. Sci. École Norm. Sup. (4) 33 (2000), no. 6, 837--934.

The state of the art for quantum groups in the locally compact setting. The introduction is very readable; the rest of the paper less so. - J. Kustermans, S. Vaes, Locally compact quantum groups in the von Neumann algebraic setting
[Leeds],
Math. Scand. 92 (2003), no. 1, 68--92.

The same idea, but working in the von Neumann algebra setting: probably easier to understand, if you are willing to go with the flow and ignore the minefield of different topologies which occur on von Neumann algebras. arXiv:math/0005219v1 - A. Van Daele, Locally compact quantum groups: the von Neumann algebra versus the $C\sp *$-algebra approach
[Leeds],
Bull. Kerala Math. Assoc. 2005, Special Issue, 153--177 (2007).

A more self-contained version of the above, essentially. arXiv:math/0602212v1 - A. Van Daele, Quantum groups with invariant integrals
[Leeds],
Proc. Natl. Acad. Sci. USA
[Leeds]
97 (2000), no. 2, 541--546 (electronic).

A very readable account of the Vaes and Kustermans work. Well worth reading.