Good text on quantum groups.

Question

I'm interested in learning about quantum groups about a $C^*$-algebraic perspective.

I'm familiar with (the basics) of topology, abstract algebra, measure theory, functional analysis (in particular $C^*$-algebras) and category theory.

However I don't know much about related topics like Hopf-algebras etc.

What references can you recommend and in what order should I read them?

Thanks!

Answer 1

If you have never seen anything about Hopf algebras I recommend perhaps looking at Section 2.2 of my own thesis. It is a very leisurely introduction in the technically easy finite dimensional case.

Perhaps for a first look at $\mathrm{C}^*$-algebraic quantum groups these notes of Roland Vergnioux might be a good idea:

• Haar integrals on finite and compact quantum group

These notes really well-motivate the definition and relate the definition very well to the commutative situation.

An overarching reference might be: - Thomas Timmermann, An Invitation to Quantum Groups and Duality - From Hopf Algebras to Multiplicative Unitaries and Beyond

However perhaps use this as a reference and instead look at graduate lecture notes such as (in no particular order):

• Teo Banica, Free Quantum Groups and Related Topics
• Adam Skalski, Quantum Symmetry Groups and Related Topics
• Amaury Freslon, Introduction to compact matrix quantum groups and their combinatorics
• Moritz Weber, Introduction to compact (matrix) quantum groups and Banica–Speicher (easy) quantum groups
• Uwe Franz, Adam Skalski, Piotr Soltan, Introduction to compact and discrete quantum groups

Between these you are in good nick.