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PhD seminar by Therese Søby Andersen (QGM)

Title: Quantum groups and their representations

2013.11.19 | Christine Dilling

Date Fri 07 May
Time 14:15 15:15
Location Aud. D3


Quantum groups were introduced around 1985 by Drinfel'd and Jimbo and were originally used to construct solutions of the so-called quantum Yang-Baxter equations. Since then, they have found numerous other applications in areas such as theoretical physics, knot theory, symplectic geometry and representations of algebraic groups.

A quantum group is a parametrised family of algebras and the complexity of its representation theory depends on whether or not the parameter q is a root of unity. If q is not a root of unity, the theory is nice and well-understood. If q is a root of unity, the theory is much more complicated and contains lots of open questions.

In this talk, I will define quantum groups from scratch and describe their representation theory. The main focus will be the non-root of unity case. Natural questions to ask in this setting are "What do the irreducible representations look like?", "Can I write every representation as a direct sum of irreducible ones?" and "How do I decompose tensor products of representations?". These questions have rather nice and simple answers that can be obtained by using the theory of weights, which is a simple but very efficient tool in representation theory.

If time permits, I will go on to treat the harder and more interesting theory that applies when q is a root of unity.