Compact quantum Lie groups

Quantum groups, which has been a major overarching theme across various branches of mathematics since late 20th century, appear in many ways. Deformation of compact Lie groups is a particularly fruitful paradigm that sits in the intersection between operator algebraic approach to quantized spaces on the one hand, and more algebraic one arising from study of quantum integrable systems on the other.
On the side of operator algebra, Woronowicz defined the C*-bialgebra representing quantized SU(2) based on his theory of pseudospaces. This gives a (noncommutative) C*-algebra of "continuous functions" on the quantized group SUq(2).
Its algebraic counterpart is the quantized universal enveloping algebra Uq(??2), due to Kulish–Reshetikhin and Sklyanin, coming from a search of algebraic structures on solutions of the Yang-Baxter equation. This is (an essentially unique) deformation of the universal enveloping algebra U(??2) as a Hopf algebra.
These structures are in certain duality, and have far-reaching generalization to compact simple Lie groups like SU(n). The interaction of ideas from both fields led to interesting results beyond original settings of these theories.
In this introductory talk, I will explain the basic quantization scheme underlying this "q-deformation", and basic properties of the associated C*-algebras. As part of more recent and advanced topics, I also want to explain an interesting relation to complex simple Lie groups through the idea of quantum double.