Drinfeld-type presentations of loop algebras
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- by Fulin Chen, Naihuan Jing, Fei Kong and Shaobin Tan PDF
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Abstract:
Let $\mathfrak {g}$ be the derived subalgebra of a Kac-Moody Lie algebra of finite-type or affine-type, let $\mu$ be a diagram automorphism of $\mathfrak {g}$, and let $\mathcal {L}(\mathfrak {g},\mu )$ be the loop algebra of $\mathfrak {g}$ associated to $\mu$. In this paper, by using the vertex algebra technique, we provide a general construction of current-type presentations for the universal central extension $\widehat {\mathfrak {g}}[\mu ]$ of $\mathcal {L}(\mathfrak {g},\mu )$. The construction contains the classical limit of Drinfeld’s new realization for (twisted and untwisted) quantum affine algebras [Soviet Math. Dokl. 36 (1988), pp. 212–216] and the Moody-Rao-Yokonuma presentation for toroidal Lie algebras [Geom. Dedicata 35 (1990), pp. 283–307] as special examples. As an application, when $\mathfrak {g}$ is of simply-laced-type, we prove that the classical limit of the $\mu$-twisted quantum affinization of the quantum Kac-Moody algebra associated to $\mathfrak {g}$ introduced in [J. Math. Phys. 59 (2018), 081701] is the universal enveloping algebra of $\widehat {\mathfrak {g}}[\mu ]$.References
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Additional Information
- Fulin Chen
- Affiliation: School of Mathematical Sciences, Xiamen University, Xiamen 361005, People’s Republic of China
- Email: chenf@xmu.edu.cn
- Naihuan Jing
- Affiliation: Department of Mathematics, North Carolina State University, Raleigh, North Carolina 27695
- MR Author ID: 232836
- Email: jing@math.ncsu.edu
- Fei Kong
- Affiliation: Key Laboratory of Computing and Stochastic Mathematics (Ministry of Education), School of Mathematics and Statistics, Hunan Normal University, Changsha 410081, People’s Republic of China
- Email: kongmath@hunnu.edu.cn
- Shaobin Tan
- Affiliation: School of Mathematical Sciences, Xiamen University, Xiamen 361005, People’s Republic of China
- Email: tans@xmu.edu.cn
- Received by editor(s): June 16, 2019
- Received by editor(s) in revised form: December 16, 2019
- Published electronically: September 14, 2020
- Additional Notes: The first author was partially supported by the Fundamental Research Funds for the Central Universities (No.20720190069) and NSF of China (No.11971397).
The second author was partially supported by NSF of China (No.11531004), Simons Foundation (No.523868), and the Humboldt Foundation.
The third author was partially supported by NSF of China (No.11701183).
The fourth author was partially supported by NSF of China (No.11531004). - © Copyright 2020 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 373 (2020), 7713-7753
- MSC (2010): Primary 17B65, 17B69
- DOI: https://doi.org/10.1090/tran/8120
- MathSciNet review: 4169672