#### 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