Pivotal Categories, Planar Algebras and Subfactors 1

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In this talk I'll define pivotal categories, and describe the diagrams we use to do calculations. The Temperley-Lieb algebra provides the first example. I'll show how the representation theory of a quantum group, and the standard invariant of a subfactor, provide two interesting generalisations of this example. In particular, I'll give an alternative description of the planar algebra structure of a subfactor, almost from scratch, in the language of pivotal categories. There'll be the bare minimum of analysis; consider this an advertisement of subfactors to representation theorists.

In the subsequent talk I'll explain how to produce generators and relations for the representations of U_q(sl_n) as a pivotal category, extending previous results for sl_2 and sl_3.