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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality 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 1.43.

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The super $\mathcal {W}_{1+\infty }$ algebra with integral central charge
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by Thomas Creutzig and Andrew R. Linshaw PDF
Trans. Amer. Math. Soc. 367 (2015), 5521-5551 Request permission

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 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