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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Tensor decomposition, parafermions, level-rank duality, and reciprocity law for vertex operator algebras
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by Cuipo Jiang and Zongzhu Lin PDF
Trans. Amer. Math. Soc. 375 (2022), 8325-8352 Request permission


For a semisimple Lie algebra $\frak {sl}_n$, the basic representation $L_{\widehat {\frak {sl}_{n}}}(1,0)$ of the affine Lie algebra $\widehat {\frak {sl}_{n}}$ is a lattice vertex operator algebra. The first main result of the paper is to prove that the commutant vertex operator algebra of $L_{\widehat {\frak {sl}_{n}}}(l,0)$ in the $l$-fold tensor product $L_{\widehat {\frak {sl}_{n}}}(1,0)^{\otimes l}$ is isomorphic to the parafermion vertex operator algebra $K(\frak {sl}_{l},n)$, which is the commutant of the Heisenberg vertex operator algebra $L_{\widehat {\frak {h}}}(n,0)$ in $L_{\widehat {\frak {sl}_l}}(n,0)$. The result provides a version of level-rank duality. The second main result of the paper is to prove more general version of the first result that the commutant of $L_{\widehat {\frak {sl}_{n}}}(l_1+\cdots +l_s, 0)$ in $L_{\widehat {\frak {sl}_{n}}}(l_1,0)\otimes \cdots \otimes L_{\widehat {\frak {sl}_{n}}}(l_s, 0)$ is isomorphic to the commutant of the vertex operator algebra generated by a Levi Lie subalgebra of $\frak {sl}_{l_1+\cdots +l_s}$ corresponding to the composition $(l_1, \cdots , l_s)$ in the rational vertex operator algebra $L_{\widehat {\frak {sl}}_{l_1+\cdots +l_s}}(n,0)$. This general version also resembles a version of reciprocity law discussed by Howe in the context of reductive Lie groups. In the course of the proof of the main results, certain Howe duality pairs also appear in the context of vertex operator algebras.
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Additional Information
  • Cuipo Jiang
  • Affiliation: School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China
  • Email:
  • Zongzhu Lin
  • Affiliation: Department of Mathematics, Kansas State University, Manhattan, Kansas 66506
  • MR Author ID: 214053
  • Email:
  • Received by editor(s): March 31, 2019
  • Received by editor(s) in revised form: January 5, 2020
  • Published electronically: October 3, 2022
  • Additional Notes: The first author was supported by China NSF grants 12171312, 11351004, 11771281
  • © Copyright 2022 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 375 (2022), 8325-8352
  • MSC (2020): Primary 17B69
  • DOI: