Topics in Geometric Representation Theory II

Part I: Representations of Yangians

Last term, we studied the abstract categorical structure on the category of quantum group modules. This term, we will be more explicit: for modules over quantum groups, affine quantum groups and Yangians, we'll study:

• Their structure as highest weight categories: introducing standard and costandard objects, Weyl and dual Weyl modules.
• Drinfeld polynomials and q-characters. ℓ-weights, the ℓ-root lattice and blocks. The meromorphic tensor structure on these categories and links to configuration spaces of points.
• The geometric interpretation of these structures on module categories, due to Nakajima.
• Category O for quantum groups attached to general Kac–Moody algebras, following Hernandez–Jimbo. Links with toroidal algebras and affine Yangians.

Reading: Chari–Pressley, A Guide to Quantum Groups (1994); Molev, Yangians and Classical Lie Algebras (2007); Nakajima, Quiver varieties and finite dimensional representations of quantum affine algebras (2001); Hernandez–Jimbo, Asymptotic representations and Drinfeld rational fractions (2012).