## Translation for finite $W$-algebras

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- by Simon M. Goodwin
- Represent. Theory
**15**(2011), 307-346 - DOI: https://doi.org/10.1090/S1088-4165-2011-00388-5
- Published electronically: April 5, 2011
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## Abstract:

A finite $W$-algebra $U(\mathfrak {g},e)$ is a certain finitely generated algebra that can be viewed as the enveloping algebra of the Slodowy slice to the adjoint orbit of a nilpotent element $e$ of a complex reductive Lie algebra $\mathfrak {g}$. It is possible to give the tensor product of a $U(\mathfrak {g},e)$-module with a finite dimensional $U(\mathfrak {g})$-module the structure of a $U(\mathfrak {g},e)$-module; we refer to such tensor products as translations. In this paper, we present a number of fundamental properties of these translations, which are expected to be of importance in understanding the representation theory of $U(\mathfrak {g},e)$.## References

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## Bibliographic Information

**Simon M. Goodwin**- Affiliation: School of Mathematics, University of Birmingham, Birmingham, B15 2TT, United Kingdom
- MR Author ID: 734259
- Email: goodwin@for.mat.bham.ac.uk
- Received by editor(s): September 22, 2009
- Received by editor(s) in revised form: June 4, 2010
- Published electronically: April 5, 2011
- © Copyright 2011
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Represent. Theory
**15**(2011), 307-346 - MSC (2010): Primary 17B10, 17B35
- DOI: https://doi.org/10.1090/S1088-4165-2011-00388-5
- MathSciNet review: 2788896