REPRESENTATION THEORY AND RELATED TOPICS SEMINAR

ABSTRACT: Arguably, representation theory consists of the theory of maps of arbitrary rings to matrix rings; this will, of course, only be interesting if one chooses interesting rings to take maps from.  Of course, for many of us, the most interesting such ring is the universal enveloping algebra of a simple Lie algebra.

In recent years, it's been noticed that such universal enveloping algebras are not sui generis; they have "brothers from another mother" constructed by quantizing symplectic singularities other than the nilcone. I'll talk about the simplest of these: the hypertoric enveloping algebra, which is constructed by taking the invariants of a torus acting on the differential operators on a vector space.

A lot of beautiful combinatorics appears in the representation theory of these algebras, including some interesting connections to the theory of hyperplanes.  There are also a lot of interesting parallels with the representation theory of Lie algebras, but with enough twists to keep some suspense in the air.