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Representation Theory

Published by the American Mathematical Society, the Representation Theory (ERT) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.7.

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Quantisation and nilpotent limits of Mishchenko–Fomenko subalgebras
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by Alexander Molev and Oksana Yakimova PDF
Represent. Theory 23 (2019), 350-378 Request permission

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
  • © Copyright 2019 American Mathematical Society
  • Journal: Represent. Theory 23 (2019), 350-378
  • MSC (2010): Primary 17B20, 17B35, 17B63, 17B80, 20G05
  • DOI: https://doi.org/10.1090/ert/531
  • MathSciNet review: 4013116