Quantum toroidal algebra

introduction

• Toroidal Lie algebras are generalizations of affine Lie algebras.
• In 1990, Moody, Rao and Yokonuma gave a presentation for untwisted toroidal Lie algebras.

quantum toroidal algebra

• parallel generator and perpendicular generator
• Miki's automorphism

expositions

• Jimbo, Representations of quantum toroidal algebras

articles

• Naihuan Jing, Chad R. Mangum, Kailash C. Misra, On realization of some twisted toroidal Lie algebras, arXiv:1605.05300 [math.RT], May 17 2016, http://arxiv.org/abs/1605.05300
• Naihuan Jing, Honglian Zhang, Hopf algebraic structures of quantum toroidal algebras, arXiv:1604.05416 [math.QA], April 19 2016, http://arxiv.org/abs/1604.05416
• B. Feigin, M. Jimbo, T. Miwa, E. Mukhin, Finite type modules and Bethe Ansatz for quantum toroidal gl(1), http://arxiv.org/abs/1603.02765v1
• Rao, S. Eswara, and Punita Batra. “On Integrable Modules for the Twisted Full Toroidal Lie Algebra.” arXiv:1509.02759 [math], September 9, 2015. http://arxiv.org/abs/1509.02759.
• Jing, Naihuan, and Chunhua Wang. “Modules of the Toroidal Lie Algebra \(\widehat{\widehat{\mathfrak{sl}}}_{2}\).” arXiv:1508.07392 [math], August 28, 2015. http://arxiv.org/abs/1508.07392.
• Feigin, B., M. Jimbo, T. Miwa, and E. Mukhin. “Branching Rules for Quantum Toroidal Gl(n).” arXiv:1309.2147 [math-Ph], September 9, 2013. http://arxiv.org/abs/1309.2147.
• Feigin, B., M. Jimbo, T. Miwa, and E. Mukhin. “Representations of Quantum Toroidal gl(n).” Journal of Algebra 380 (April 15, 2013): 78–108. doi:10.1016/j.jalgebra.2012.12.029.
• Hernandez, David. “Quantum Toroidal Algebras and Their Representations.” arXiv:0801.2397 [math], January 15, 2008. http://arxiv.org/abs/0801.2397.
• Miki, Kei. “Quantum Toroidal Algebra Uq(sl2,tor) and R Matrices.” Journal of Mathematical Physics 42, no. 5 (May 1, 2001): 2293–2308. doi:10.1063/1.1357198.
• Takemura, K., and D. Uglov. “Representations of the Quantum Toroidal Algebra on Highest Weight Modules of the Quantum Affine Algebra of Type gl(N).” arXiv:math/9806134, June 24, 1998. http://arxiv.org/abs/math/9806134.
• Saito, Yoshihisa. "Quantum toroidal algebras and their vertex representations." Publications of the Research Institute for Mathematical Sciences 34.2 (1998): 155-177.