Paul Smith: The quantum group SL

Current Topics Seminar

Speaker	Paul Smith
Title	The quantum group SL_q(2)
Date	April 17 4:00pm PDL C-36

Quantum groups are not groups, but they have a representation theory. In fact, they behave very much like groups, and in some sense are deformations of groups. For example, there is a quantum group corresponding to each semisimple Lie group and just as the representation theory of a Lie group is intimately related to the geometry of spaces on which it acts, so is the representation theory of a quantum group. The difference is that the "spaces" on which a quantum group acts are "non-commutative". I will explain what is meant by this. In this talk we will focus on the simplest example, the quantum-group analogue of the 2x2 complex matrices with determinant one. We will outline the main ideas needed to prove that the ordinary Riemann sphere is a "homogenous space" for SL_q(2). Roughly speaking, SL_q(2) has a "Borel subgroup" B_q and the category of B_q-equivariant sheaves on SL_q(2) is equivalent to the category of quasi-coherent sheaves on the complex projective line, the Riemann sphere.