# Seminar : Quantum affine algebras and spectra of quantum integrable systems

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This is the page for the seminar on QFT at the University of Queensland. We focus on Quantum Affine Algebras for Semester 2, 2015. The main goal is to understand various aspects of the theory of quantum affine algebras in mathematical physics.

**when**: Thursdays 3-4:30 pm**where**: Priestly Building Seminar Room 67-442

## Contents

- 1 topics
- 1.1 Quantum inverse scattering method (6/8/2015, Jon Links)
- 1.2 introduction (13/8/2015, Chul-hee Lee)
- 1.3 TQ relation of the eight-vertex model (20/8/2015, Ole Warnaar)
- 1.4 finite-dimensional representations of affine Lie algebras (Masoud Kamgarpour)
- 1.5 quantum affine algebras and its realizations (Inna Lukyanenko)
- 1.6 finite-dimensional representations of quantum affine algebras (Sean Michael Wilson)
- 1.7 theory of q-characters (Chul-hee Lee)
- 1.8 video session
- 1.9 constuction of transfer-matrices (Chul-hee Lee)
- 1.10 Borel algebras and their representations (Peter McNamara)
- 1.11 TQ relations from representation theoretic point of view
- 1.12 optional topics

- 2 links
- 3 references

## topics

Some references are given for each topic. But **we won't necessarily follow them strictly or cover them all**.

### Quantum inverse scattering method (6/8/2015, Jon Links)

- TQ relations from algebraic Bethe ansatz

### introduction (13/8/2015, Chul-hee Lee)

- overview
- distribution of the remaining talks

### TQ relation of the eight-vertex model (20/8/2015, Ole Warnaar)

- Chapter 10 of [Baxter 1982]
- Baxter, Rodney J. “Partition Function of the Eight-Vertex Lattice Model.” Annals of Physics 70 (1972): 193–228.

### finite-dimensional representations of affine Lie algebras (Masoud Kamgarpour)

- Chari, Vyjayanthi. “Representations of Affine and Toroidal Lie Algebras.” arXiv:1009.1336 [math], September 7, 2010. http://arxiv.org/abs/1009.1336.
- Rao, S. Eswara. “On Representations of Loop Algebras.” Communications in Algebra 21, no. 6 (1993): 2131–53. doi:10.1080/00927879308824668.
- Senesi, Prasad. “Finite-Dimensional Representation Theory of Loop Algebras: A Survey.” arXiv:0906.0099 [math], May 30, 2009. http://arxiv.org/abs/0906.0099.

### quantum affine algebras and its realizations (Inna Lukyanenko)

- Lecture 6 of [EFK 1998]
- Drinfel'd, V. G. “A New Realization of Yangians and of Quantum Affine Algebras.” Doklady Akademii Nauk SSSR 296, no. 1 (1987): 13–17.
- Jimbo, Michio. “Aq-Difference Analogue of U(g) and the Yang-Baxter Equation.” Letters in Mathematical Physics 10, no. 1 (July 1985): 63–69. doi:10.1007/BF00704588.
- Drinfeld, Vladimir G. "Quantum groups." (1986): 789-820. http://www.mathunion.org/ICM/ICM1986.1/Main/icm1986.1.0798.0820.ocr.pdf

### finite-dimensional representations of quantum affine algebras (Sean Michael Wilson)

- Chapter 12 of [CP 1995]
- Chari, Vyjayanthi, and Andrew Pressley. “Quantum Affine Algebras and Their Representations.” In Representations of Groups (Banff, AB, 1994), 16:59–78. CMS Conf. Proc. Amer. Math. Soc., Providence, RI, 1995. http://www.ams.org/mathscinet-getitem?mr=1357195.

### theory of q-characters (Chul-hee Lee)

- Frenkel, Edward, and Nicolai Reshetikhin. 1999. “The $q$-characters of Representations of Quantum Affine Algebras and Deformations of $\scr W$-algebras.” In Recent Developments in Quantum Affine Algebras and Related Topics (Raleigh, NC, 1998), 248:163–205. Contemp. Math. Providence, RI: Amer. Math. Soc. http://www.ams.org/mathscinet-getitem?mr=1745260. http://arxiv.org/abs/math/9810055.
- Knight, Harold. 1995. “Spectra of Tensor Products of Finite-Dimensional Representations of Yangians.” Journal of Algebra 174 (1): 187–196. doi:10.1006/jabr.1995.1123.

### video session

- Frenkel, Baxter relations and polynomiality of transfer-matrices http://video.impa.br/index.php?page=symmetries-in-mathematics-and-physics-ii, video lecture recorded on 25/06/2013

### constuction of transfer-matrices (Chul-hee Lee)

- Frenkel, Edward, and Nicolai Reshetikhin. 1999. “The $q$-characters of Representations of Quantum Affine Algebras and Deformations of $\scr W$-algebras.” In Recent Developments in Quantum Affine Algebras and Related Topics (Raleigh, NC, 1998), 248:163–205. Contemp. Math. Providence, RI: Amer. Math. Soc. http://www.ams.org/mathscinet-getitem?mr=1745260. http://arxiv.org/abs/math/9810055.

### Borel algebras and their representations (Peter McNamara)

- Hernandez, David, and Michio Jimbo. “Asymptotic Representations and Drinfeld Rational Fractions.” Compositio Mathematica 148, no. 5 (2012): 1593–1623. doi:10.1112/S0010437X12000267. http://www.ams.org/mathscinet-getitem?mr=2982441
- Bazhanov, Vladimir V., Sergei L. Lukyanov, and Alexander B. Zamolodchikov. “Integrable Structure of Conformal Field Theory. III. The Yang-Baxter Relation.” Communications in Mathematical Physics 200, no. 2 (1999): 297–324. doi:10.1007/s002200050531.

### TQ relations from representation theoretic point of view

- Frenkel, Edward, and Nicolai Reshetikhin. 1999. “The $q$-characters of Representations of Quantum Affine Algebras and Deformations of $\scr W$-algebras.” In Recent Developments in Quantum Affine Algebras and Related Topics (Raleigh, NC, 1998), 248:163–205. Contemp. Math. Providence, RI: Amer. Math. Soc. http://www.ams.org/mathscinet-getitem?mr=1745260. http://arxiv.org/abs/math/9810055.
- Frenkel, Edward, and David Hernandez. 2013. “Baxter’s Relations and Spectra of Quantum Integrable Models”. ArXiv e-print 1308.3444. http://arxiv.org/abs/1308.3444.

### optional topics

- quantum affine sl(2) and its finite-dimensional representations
- Lecture 9 of [EFK 1998]

- universal R-matrix

## links

## references

<HarvardReferences>

- [*EFK 1998] Etingof, Pavel I., Igor B. Frenkel, and Alexander A. Kirillov Jr. Lectures on Representation Theory and Knizhnik-Zamolodchikov Equations. Vol. 58. Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 1998. http://www.ams.org/mathscinet-getitem?mr=1629472.
- [*CP 1995] Chari, Vyjayanthi, and Andrew Pressley. A Guide to Quantum Groups. Cambridge University Press, Cambridge, 1995. http://www.ams.org/mathscinet-getitem?mr=1358358.
- [*Baxter 1982] R. J. Baxter, "Exactly Solved Models in Statistical Mechanics" Academic Press, 1982 http://www.ams.org/mathscinet-getitem?mr=998375. http://physics.anu.edu.au/theophys/baxter_book.php

</HarvardReferences>