The super $\mathcal {W}_{1+\infty }$ algebra with integral central charge
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- by Thomas Creutzig and Andrew R. Linshaw PDF
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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.References
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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
- 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. 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
- © Copyright 2015 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 367 (2015), 5521-5551
- MSC (2010): Primary 17B69
- DOI: https://doi.org/10.1090/S0002-9947-2015-06214-X
- MathSciNet review: 3347182