Topics in Representation Theory
Course information
- Time and Location: Tuesdays and Thursdays 10.10-11.25AM (622 Math)
- Office hours: Tuesdays and Thursdays 3-4PM (413 Math)
- Textbook. This course will be based entirely on lecture notes.
List of topics and references
Lecture notes
- Lecture 0
Lattice models of statistical mechanics; Derivation of
Yang-Baxter equation; RTT algebras.
- Lecture 1
Kac-Moody algebras. Root systems.
- Lecture 2
Finite and affine Lie algebras. Braid groups.
- Lecture 3
Quasi-triangular Hopf algebras.
- Lecture 4
Quantum groups associated to Kac-Moody groups.
R-matrix.
- Lecture 5
Quantum groups continued. Serre relations. Category O.
Integrable representations.
- Lecture 6
Quantum Weyl group. Braid relations.
- Lecture 7
Fusion operator and dynamical (quantum) Weyl group.
- Lecture 8
Quantum affine algebras. Drinfeld's new presentation.
- Lecture 9
Quantum loop algebras. Classification of irreducible
finite-dimensional representations (Drinfeld polynomials) - statement without a proof.
- Lecture 10
R-matrix of quantum loop algebras. Spectral dependence.
Convergence argument of Etingof and Moura. Introduction
to meromorphic tensor categories.
- Lecture 11
Drinfeld coproduct as a meromorphic tensor product.
Soibelman's definition of meromorphic braided tensor categories.
- Lecture 12
Meromorphic tensor categories continued. Braiding
on f.d. representations of quantum loop algebras, relative to
Drinfeld coproduct.
- Lecture 13
q-difference equations. Their monodromy and doubly-quasi-periodic
functions. Jacobi's theta function.
- Lecture 14
Definition of Yangians. Drinfeld's new presentation.
Levendorskii's presentation.
- Lecture 15
Classification of irreducible finite-dimensional representations
of Yangians - reduction to rank 1.
- Lecture 16
Classification of irreducible finite-dimensional representations
of Yangians - proof for rank 1.
- Lecture 17
Representation theory of Yangians continued: Knight's lemma.
q-characters. Drinfeld's R-matrix (statement only).
- Lecture 18
Drinfeld tensor product (mero.) for f.d. representations
of Yangians. Construction of the corresponding braiding.
Sliced subcategories.
- Lecture 19
Additive difference equations. Euler's Gamma function.
- Lecture 20
From representations of Yangians to those of quantum loop algebras
using additive difference equations.