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Transactions of the American Mathematical Society

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

 
 

 

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 $\mathfrak {g} \mathfrak {l}(n|n)$. 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 85Β Ave., Edmonton, AB T6G 2R3, Canada
MR Author ID: 832147
ORCID: 0000-0002-7004-6472
Email: creutzig@ualberta.ca

Andrew R. Linshaw
Affiliation: Department of Mathematics, University of Denver, 2199 S.Β University Blvd., Denver, Colorado 80208
MR Author ID: 791304
Email: andrew.linshaw@du.edu

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