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Representation Theory

Published by the American Mathematical Society since 1997, this electronic-only journal is devoted to research in representation theory and seeks to maintain a high standard for exposition as well as for mathematical content. All articles are freely available to all readers and with no publishing fees for authors.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.71.

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Quantisation and nilpotent limits of Mishchenko–Fomenko subalgebras
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by Alexander Molev and Oksana Yakimova
Represent. Theory 23 (2019), 350-378
Published electronically: September 30, 2019


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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Bibliographic Information
  • Alexander Molev
  • Affiliation: School of Mathematics and Statistics, University of Sydney, New South Wales 2006, Australia
  • MR Author ID: 207046
  • Email:
  • 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:
  • 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:
  • MathSciNet review: 4013116