#### Vol. 7, No. 8, 2013

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Principal $W$-algebras for $\operatorname{GL}(m\vert n)$

### Jonathan Brown, Jonathan Brundan and Simon M. Goodwin

Vol. 7 (2013), No. 8, 1849–1882
##### Abstract

We consider the (finite) $W$-algebra ${W}_{m|n}$ attached to the principal nilpotent orbit in the general linear Lie superalgebra ${\mathfrak{g}\mathfrak{l}}_{m|n}\left(ℂ\right)$. Our main result gives an explicit description of ${W}_{m|n}$ as a certain truncation of a shifted version of the Yangian $Y\left({\mathfrak{g}\mathfrak{l}}_{1|1}\right)$. We also show that ${W}_{m|n}$ admits a triangular decomposition and construct its irreducible representations.

##### Keywords
$W$-algebras, Lie superalgebras
Primary: 17B10
Secondary: 17B37
##### Milestones
Received: 10 May 2012
Accepted: 17 December 2012
Published: 24 November 2013
##### Authors
 Jonathan Brown Department of Mathematics, Computer Science, and Statistics State University of New York College at Oneonta Oneonta, NY 13820 United States Jonathan Brundan Department of Mathematics University of Oregon Eugene, OR 97403 United States Simon M. Goodwin School of Mathematics University of Birmingham Birmingham  B15 2TT United Kingdom