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The super $ \mathcal{W}_{1+\infty}$ algebra with integral central charge


Authors: Thomas Creutzig and Andrew R. Linshaw
Journal: Trans. Amer. Math. Soc. 367 (2015), 5521-5551
MSC (2010): Primary 17B69
DOI: https://doi.org/10.1090/S0002-9947-2015-06214-X
Published electronically: February 3, 2015
MathSciNet review: 3347182
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Abstract: The Lie superalgebra $ \mathcal {S}\mathcal {D}$ of regular differential operators on the super circle has a universal central extension $ \widehat {\mathcal {S}\mathcal {D}}$. For each $ c\in \mathbb{C}$, the vacuum module $ \mathcal {M}_c(\widehat {\mathcal {S}\mathcal {D}})$ of central charge $ c$ admits a vertex superalgebra structure, and $ \mathcal {M}_c(\widehat {\mathcal {S}\mathcal {D}}) \cong \mathcal {M}_{-c}(\widehat {\mathcal {S}\mathcal {D}})$. The irreducible quotient $ \mathcal {V}_c(\widehat {\mathcal {S}\mathcal {D}})$ of the vacuum module is known as the super $ \mathcal {W}_{1+\infty }$ algebra. We show that for each integer $ n>0$, $ \mathcal {V}_{n}(\widehat {\mathcal {S}\mathcal {D}})$ has a minimal strong generating set consisting of $ 4n$ fields, and we identify it with a $ \mathcal {W}$-algebra associated to the purely odd simple root system of $ \mathfrak{g} \mathfrak{l}(n\vert n)$. Finally, we realize $ \mathcal {V}_{n}(\widehat {\mathcal {S}\mathcal {D}})$ as the limit of a family of commutant vertex algebras that generically have the same graded character and possess a minimal strong generating set of the same cardinality.


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Additional Information

Thomas Creutzig
Affiliation: Department of Mathematics, University of Alberta, 116 St. and 85 Ave., Edmonton, AB T6G 2R3, Canada
Email: creutzig@ualberta.ca

Andrew R. Linshaw
Affiliation: Department of Mathematics, University of Denver, 2199 S. University Blvd., Denver, Colorado 80208
Email: andrew.linshaw@du.edu

DOI: https://doi.org/10.1090/S0002-9947-2015-06214-X
Received by editor(s): September 28, 2012
Received by editor(s) in revised form: June 10, 2013
Published electronically: February 3, 2015
Article copyright: © Copyright 2015 American Mathematical Society

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