MAT 298 (Reading course in Representation Theory) Spring 2020
MAT 298 (Reading course in Representation Theory) Spring 2020
This is a continuation of the reading course from Winter 2020. We meet on Tuesdays at 11am, send me an email if you want to join via Zoom. In this page, I will be posting the course schedule, as well as exercise sheets.
Schedule:
Tuesday, March 31st. José introduced Lie algebras, proved Schur-Weyl duality for the Lie algebra gl_n and introduced universal enveloping algebras, as well as a sketch of a proof of the PBW theorem. Notes.
Tuesday, April 7. Eugene introduced the Coxeter presentation of Sn, (pure) braid groups and Hecke algebras, computed the dimension of the Hecke algebras and finished with a proof that for generic values of q the Hecke algebra is semisimple. Notes.
Tuesday, April 14. Yue spoke about Temperley-Lieb algebras: their diagrammatic presentation, dimension, and presentation by generators and relations. She also related Hecke algebras to Temperley-Lieb algebras, and computed the kernel of the explicit epi from the former to the latter. Notes.
Tuesday, April 21. Milo solved problems on sl(2).
Tuesday, April 28. Daniel solved more problems on sl(2).
Tuesday, Cinco de Mayo. José introduced Hopf algebras and defined quantum sl(2). Notes.
Tuesday, May 12. José elaborated on the connection between classical and quantum sl(2), and constructed the universal R-matrix for sl(2). Notes.
Tuesday, May 19. Eugene spoke about cabling identities and the Yang-Baxter equation, and how this follows automatically using the Drinfeld double construction of the R-matrix. Notes.
Tuesday, May 26. Alex solved exercises on quantum sl(2). Notes.
Tuesday, June 2. Eugene spoke about Witten-Reshetikhin-Turaev invariants for links from quantum groups and in particular the (colored) Jones polynomials associated to quantum sl(2). Notes.