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Quantisation and nilpotent limits of Mishchenko–Fomenko subalgebras


Authors: Alexander Molev and Oksana Yakimova
Journal: Represent. Theory 23 (2019), 350-378
MSC (2010): Primary 17B20, 17B35, 17B63, 17B80, 20G05
DOI: https://doi.org/10.1090/ert/531
Published electronically: September 30, 2019
MathSciNet review: 4013116
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Abstract: For any simple Lie algebra $\mathfrak {g}$ and an element $\mu \in \mathfrak {g}^*$, the corresponding commutative subalgebra $\mathcal {A}_{\mu }$ of $\mathcal {U}(\mathfrak {g})$ is defined as a homomorphic image of the Feigin–Frenkel centre associated with $\mathfrak {g}$. It is known that when $\mu$ is regular this subalgebra solves Vinberg’s quantisation problem, as the graded image of $\mathcal {A}_{\mu }$ coincides with the Mishchenko–Fomenko subalgebra $\overline {\mathcal {A}}_{\mu }$ of $\mathcal {S}(\mathfrak {g})$. By a conjecture of Feigin, Frenkel, and Toledano Laredo, this property extends to an arbitrary element $\mu$. We give sufficient conditions on $\mu$ which imply the property. In particular, this proves the conjecture in type C and gives a new proof in type A. We show that the algebra $\mathcal {A}_{\mu }$ is free in both cases and produce its generators in an explicit form. Moreover, we prove that in all classical types generators of $\mathcal {A}_{\mu }$ can be obtained via the canonical symmetrisation map from certain generators of $\overline {\mathcal {A}}_{\mu }$. The symmetrisation map is also used to produce free generators of nilpotent limits of the algebras $\mathcal {A}_{\mu }$ and to give a positive solution of Vinberg’s problem for these limit subalgebras.


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Additional Information

Alexander Molev
Affiliation: School of Mathematics and Statistics, University of Sydney, New South Wales 2006, Australia
MR Author ID: 207046
Email: alexander.molev@sydney.edu.au

Oksana Yakimova
Affiliation: Mathematisches Institut, Universität zu Köln, Weyertal 86-90, 50931 Köln, Germany
Address at time of publication: Institut für Mathematik, Friedrich-Schiller-Universität Jena, Jena, 07737, Deutschland
MR Author ID: 695654
Email: yakimova.oksana@uni-koeln.de

Received by editor(s): December 14, 2017
Received by editor(s) in revised form: February 23, 2019
Published electronically: September 30, 2019
Additional Notes: The second author is the corresponding author.
The second author was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) — project number 330450448
The authors acknowledge the support of the Australian Research Council, grant DP150100789
Article copyright: © Copyright 2019 American Mathematical Society