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Quantized symplectic actions and $ W$-algebras


Author: Ivan Losev
Journal: J. Amer. Math. Soc. 23 (2010), 35-59
MSC (2000): Primary 17B35, 53D55
DOI: https://doi.org/10.1090/S0894-0347-09-00648-1
Published electronically: September 18, 2009
MathSciNet review: 2552248
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Abstract: With a nilpotent element in a semisimple Lie algebra $ \mathfrak{g}$ one associates a finitely generated associative algebra $ \mathcal{W}$ called a $ W$-algebra of finite type. This algebra is obtained from the universal enveloping algebra $ U(\mathfrak{g})$ by a certain Hamiltonian reduction. We observe that $ \mathcal{W}$ is the invariant algebra for an action of a reductive group $ G$ with Lie algebra $ \mathfrak{g}$ on a quantized symplectic affine variety and use this observation to study $ \mathcal{W}$. Our results include an alternative definition of $ \mathcal{W}$, a relation between the sets of prime ideals of $ \mathcal{W}$ and of the corresponding universal enveloping algebra, the existence of a one-dimensional representation of $ \mathcal{W}$ in the case of classical $ \mathfrak{g}$ and the separation of elements of $ \mathcal{W}$ by finite-dimensional representations.


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Additional Information

Ivan Losev
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139
Email: ivanlosev@math.mit.edu

DOI: https://doi.org/10.1090/S0894-0347-09-00648-1
Keywords: $W$-algebras, nilpotent elements, universal enveloping algebras, deformation quantization, prime ideals, finite-dimensional representations
Received by editor(s): August 17, 2007
Published electronically: September 18, 2009
Article copyright: © Copyright 2009 American Mathematical Society

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