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Transactions of the American Mathematical Society

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

 
 

 

Vertex algebraic intertwining operators among generalized Verma modules for $ \widehat{\mathfrak{sl}(2,\mathbb{C})}$


Authors: Robert McRae and Jinwei Yang
Journal: Trans. Amer. Math. Soc. 370 (2018), 2351-2390
MSC (2010): Primary 17B67, 17B69
DOI: https://doi.org/10.1090/tran/7012
Published electronically: November 30, 2017
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Abstract: We construct vertex algebraic intertwining operators among certain generalized Verma modules for $ \widehat {\mathfrak{sl}(2,\mathbb{C})}$ and calculate the corresponding fusion rules. Additionally, we show that under some conditions these intertwining operators descend to intertwining operators among one generalized Verma module and two (generally non-standard) irreducible modules. Our construction relies on the irreducibility of the maximal proper submodules of generalized Verma modules appearing in the Garland-Lepowsky resolutions of standard $ \widehat {\mathfrak{sl}(2,\mathbb{C})}$-modules. We prove this irreducibility using the composition factor multiplicities of irreducible modules in Verma modules for symmetrizable Kac-Moody Lie algebras of rank $ 2$, given by Rocha-Caridi and Wallach.


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Additional Information

Robert McRae
Affiliation: Beijing International Center for Mathematical Research, Peking University, Beijing 100084, People’s Republic of China
Address at time of publication: Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37240
Email: robert.h.mcrae@vanderbilt.edu

Jinwei Yang
Affiliation: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556
Address at time of publication: Department of Mathematics, Yale University, New Haven, Connecticut 06520
Email: jinwei.yang@yale.edu

DOI: https://doi.org/10.1090/tran/7012
Received by editor(s): December 14, 2015
Received by editor(s) in revised form: June 23, 2016
Published electronically: November 30, 2017
Article copyright: © Copyright 2017 American Mathematical Society

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