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Horospherical splittings of $\mathfrak g$ and related Poisson commutative subalgebras of $\mathcal S(\mathfrak g)$

Dmitri Panyushev, Oksana Yakimova

Abstract

Let a Lie algebra $\mathfrak q$ be a linear sum of two complementary subalgebras $\mathfrak h$ and $\mathfrak r$. We continue our investigations initiated in (J. London Math. Soc. 103 (2021), 1577-1595), where compatible Poisson brackets associated with splitting $\mathfrak q=\mathfrak h\oplus\mathfrak r$ and related Poisson-commutative subalgebras of the symmetric algebra $\mathcal S(\mathfrak q)$ are studied. In this article, we further develop the general theory and study in more details splittings of the reductive Lie algebras such that both $\mathfrak h$ and $\mathfrak r$ are solvable horospherical subalgebras. We also derive some results of the Adler-Kostant-Symes theory using our approach.

Horospherical splittings of $\mathfrak g$ and related Poisson commutative subalgebras of $\mathcal S(\mathfrak g)$

Abstract

Let a Lie algebra be a linear sum of two complementary subalgebras and . We continue our investigations initiated in (J. London Math. Soc. 103 (2021), 1577-1595), where compatible Poisson brackets associated with splitting and related Poisson-commutative subalgebras of the symmetric algebra are studied. In this article, we further develop the general theory and study in more details splittings of the reductive Lie algebras such that both and are solvable horospherical subalgebras. We also derive some results of the Adler-Kostant-Symes theory using our approach.
Paper Structure (20 sections, 34 theorems, 34 equations, 4 figures, 1 table)

This paper contains 20 sections, 34 theorems, 34 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries on Poisson brackets and contractions
  3. The Lenard--Magri scheme
  4. Contractions and invariants
  5. Poisson-commutative subalgebras related to splittings
  6. On the numbers $s_0$ and $s_\infty$
  7. More results on $s_0$ and $s_\infty$
  8. The non-degenerate case
  9. Horospherical contractions and splittings
  10. A necessary condition for the presence of g.g.s.
  11. A horospherical splitting of ${\mathfrak g}\dotplus{\mathfrak t}$.
  12. Semi-horospherical splittings and involutions
  13. ${\mathfrak g}=\mathfrak{sl}_{2n}$
  14. ${\mathfrak g}=\mathfrak{sl}_{2n+1}$
  15. ${\mathfrak g}={\mathfrak{so}}_{2n}$ and ${\mathfrak g}_0=\mathfrak{so}_{n+1}\oplus\mathfrak{so}_{n-1}$
  16. ...and 5 more sections

Key Result

Lemma 2.1

Let ${\mathfrak m}$ and $\tilde{{\mathfrak m}}$ be different complementary subspaces to ${\mathfrak h}$. Then there is a linear operator ${\mathcal{L}}: {\mathfrak q}\to {\mathfrak q}$ such that ${\mathcal{L}}\vert_{\mathfrak h}={\sf id}$, ${\mathcal{L}}({\mathfrak m})=\tilde{{\mathfrak m}}$, and ${

Figures (4)

  • Figure 1: The Satake diagram for $(\mathfrak{sl}_{2n}, {\mathfrak{sl}}_n\dotplus{\mathfrak{sl}}_n\dotplus\Bbbk)$
  • Figure 2: The Satake diagram for $({\mathfrak{so}}_{2n}, \mathfrak{so}_{n+1}\dotplus\mathfrak{so}_{n-1})$
  • Figure 3: The Satake diagram for $(\EuScript E_6, \mathfrak{sl}_6 \dotplus \mathfrak{sl}_2 )$
  • Figure :

Theorems & Definitions (71)

  • Lemma 2.1
  • proof
  • Definition 1
  • Example 2.2
  • Theorem 2.3: contr
  • Remark 2.4
  • Lemma 2.5
  • proof
  • Proposition 3.1
  • proof
  • ...and 61 more