## Topology & Geometric Group Theory Seminar

## Fall 2011

### 1:30 – 2:30, Malott 203

### Thursday, September 22

**Greg
Kuperberg**, UC Davis

*Buildings, spiders, and geometric Satake*

Louis Kauffman found a special description of the Jones polynomial and
the representation theory of U_{q}(sl(2)) in which each skein
space has a basis of planar matchings. There is a similar calculus
(discovered independently by myself and the late François
Jaeger) for each of the three rank 2 simple Lie algebras
A_{2},B_{2}, and G_{2}. These skein theories,
called "spiders", can also be viewed as Gröbner-type
presentations of pivotal categories. In each of the four cases
(optionally also including the semisimple case A_{1} ×
A_{1}), the Gröbner basis property yields a basis of
skein diagrams called "webs". The basis webs are defined by an
interesting non-positive curvature condition.

I will discuss a new connection between these spiders and the
geometric Satake correspondence, which relates the representation
category of a simple Lie algebra to an affine building of the
Langlands dual algebra. In particular, any such building is CAT(0),
which seems to explain the non-positive curvature of basis webs.

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