Dynkin Automorphism Actions on Gaudin Algebras

By V. Zveryk. Transformation groups, 2023

165 citations recorded by the source corpus.

We study the action of Dynkin diagram automorphisms σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma $$\end{document} on generalized Gaudin algebras, focusing in particular on the big Gaudin algebra B(g)⊂(U(g)⊗S(g))g\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {B}(\mathfrak {g}) \subset (U(\mathfrak {g}) \otimes S(\mathfrak {g}))^{\mathfrak {g}}$$\end{document} and its evaluated versions Bλ(g)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {B}^\lambda (\mathfrak {g})$$\end{document} and Bχ(g)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {B}_\chi (\mathfrak {g})$$\end{document}. We show isomorphisms between the coinvariants of the generalized Gaudin algebras associated with g∨\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {g}^\vee $$\end{document} and the generalized Gaudin algebras associated with gσ∨\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {g}_\sigma ^\vee $$\end{document}, where gσ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak g_\sigma $$\end{document} is the fixed point subalgebra. In particular, we get an isomorphism Bλ(g∨)σ≃Bλ(gσ∨)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {B}^\lambda (\mathfrak {g}^\vee )_\sigma \simeq \mathcal {B}^\lambda (\mathfrak {g}^\vee _\sigma )$$\end{document} for any σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma $$\end{document}-invariant dominant weight λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document}, which allows us to reprove Jantzen’s twining formula. Our approach relies on interpreting generalized Gaudin algebras via spaces of opers, which explains the appearance of the Langlands duals in our results and in Jantzen’s twining formula.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/9465bd60b27a537bfa5c204ebcbd91751b2b2d08

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In Algebra and Number Theory

  1. Jürgen Fuchs
  2. Ingo Runkel
  3. Christoph Schweigert

In Geometry and Topology

  1. Jürgen Fuchs
  2. Ingo Runkel
  3. Christoph Schweigert
  4. Michael R Douglas
  5. Frederik Denef
  6. T. Bridgeland
  7. Liang Kong
  8. Yukinobu Toda

In Mathematical Physics

  1. Jürgen Fuchs
  2. Ingo Runkel
  3. Christoph Schweigert
  4. Michael R Douglas
  5. T. Bridgeland
  6. Frederik Denef
  7. Liang Kong
  8. Yukinobu Toda

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