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28/06/2011, 16:00 — 17:00 — Room P3.10, Mathematics Building

Nitu Kitchloo, *Johns Hopkins and UCSD*

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Geometry, Topology and Representation Theory of Loop Groups

In this sequence of three talks, I will aim to introduce the
algebraic and geometric structure of Loop groups and their
representations. We will begin with the basic structure of Affine
Lie algebras. This will lead us to the algebraic theory of positive
energy representations indexed by the level. On the geometric side,
we will introduce the Affine Loop group and relate it to the
central extension of the smooth loop group. We will also study the
example of the special unitary group in some detail. In the
remaining time, I will go into some of the deeper structure of Loop
groups. This includes fusion in the representations of a given
level (via the geometric notion of conformal blocks). Time
permitting, I will also describe the homotopy type of the
classifying space of Loop groups. No special background is
required. It would be helpful to know the basic theory of root
systems for semisimple Lie algebras, though this is not a strict
requirement.

#### References

- Arnaud Beauville, Conformal
blocks, fusion rules and the Verlinde formula, Proc. of the
Hirzebruch 65 Conf. on Algebraic Geometry, Israel Math. Conf. Proc.
9, 75-96 (1996).
- Victor Kac, Infinite dimensional Lie algebras, Cambridge
University Press (1990).
- Nitu Kitchloo, On the
topology of Kac-Moody groups (2008).
- Andrew Pressley and Graeme Segal, Loop Groups, Oxford
University Press (1986).

#### See also

http://www.math.ist.utl.pt/~ggranja/SummerLect11_files/Loop.pdf

http://www.math.ist.utl.pt/~ggranja/SummerLect11_files/LoopGps.pdf