# On an iceberg which Lie algebras are a tip of

SPEAKER: Ben Webster

TITLE: On an iceberg which Lie algebras are a tip of (joint w/Tom Braden, Anthony Licata and Nick Proudfoot)

ABSTRACT: I’ll show how the universal enveloping algebras of semi-simple Lie algebras are one example of a very general construction, which takes in a symplectic variety and spits out a noncommutative algebra. Other algebras which appear this way include Cherednik algebras and finite W-algebras, but by applying this to varieties such as quiver varieties and hypertoric varieties, we also obtain many examples of interesting algebras which seem not to have been previously considered. This leads to an approximately infinite supply of questions of the form: does fact X about the representation theory of Lie algebras generalize to these algebras constructed from other varieties? Interesting facts I would like to generalize include the Beilinson-Bernstein localization theorem, the classification of finite dimensional representations, the theory of primitive ideals and the structure of category O (including twisting and shuffling functors) and Koszul duality between various categories of representations. I’ll describe the baby steps my collaborators and I have taken in this direction. If I have time, I’ll also describe how the Koszul dualities mentioned give one piece of evidence for the existence of a duality between symplectic singularities.