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.

• Exercises on sl(2).

• 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.

• Exercises on quantum sl(2).

• Tuesday, May 12. José elaborated on the connection between classical and quantum sl(2), and constructed the universal R-matrix for sl(2). Notes.

• Exercises on the R-matrix for sl(2).

• 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.