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Quantum groups and their representations

Therese Søby Andersen
Fredag, 7 maj, 2010, at 14:15-15:15, in Aud. D3 (1531-215)
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.

Organiseret af: CTQM/QGM
Kontaktperson: Magnus Roed Lauridsen