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Integrable systems associated to the filtrations of Lie algebras

Bozidar Jovanovic, Tijana Sukilovic, Srdjan Vukmirovic

Abstract

In 1983 Bogoyavlenski conjectured that if the Euler equations on a Lie algebra $\mathfrak g_0$ are integrable, then their certain extensions to semisimple lie algebras $\mathfrak g$ related to the filtrations of Lie algebras $\mathfrak g_0\subset \mathfrak g_1\subset \mathfrak g_2\dots\subset\mathfrak g_{n-1}\subset \mathfrak g_n=\mathfrak g$ are integrable as well. In particular, by taking $\mathfrak g_0=\{0\}$ and natural filtrations of $\mathfrak{so}(n)$ and $\mathfrak{u}(n)$, we have Gel'fand-Cetlin integrable systems. We proved the conjecture for filtrations of compact Lie algebras $\mathfrak g$: the system are integrable in a noncommutative sense by means of polynomial integrals. Various constructions of complete commutative polynomial integrals for the system are also given.

Integrable systems associated to the filtrations of Lie algebras

Abstract

In 1983 Bogoyavlenski conjectured that if the Euler equations on a Lie algebra are integrable, then their certain extensions to semisimple lie algebras related to the filtrations of Lie algebras are integrable as well. In particular, by taking and natural filtrations of and , we have Gel'fand-Cetlin integrable systems. We proved the conjecture for filtrations of compact Lie algebras : the system are integrable in a noncommutative sense by means of polynomial integrals. Various constructions of complete commutative polynomial integrals for the system are also given.

Paper Structure

This paper contains 10 sections, 16 theorems, 105 equations.

Table of Contents

  1. Introduction
  2. Complete algebras of functions on Poisson manifolds
  3. Polynomial noncommutative integrability
  4. The problem of a polynomial commutative integrability
  5. Commutative polynomial integrals
  6. Isotropy subgroups and the Mishcenko-Fomenko shifting of argument method
  7. Mikityuk's construction with symmetric pairs and its variation
  8. Bogoyavlenski's integrals
  9. Diagonal subgroups
  10. Reyman's construction: the shifting of argument and symmetric pairs

Key Result

Lemma 1

If $f$ and $g$ Lie--Poisson commute on $\mathfrak{g}_i$, then their lifts $\tilde{f}=\mathop{\mathrm{\mathrm{pr}}}\nolimits^*_{\mathfrak{g}_i} f$ and $\tilde{g}=\mathop{\mathrm{\mathrm{pr}}}\nolimits^*_{\mathfrak{g}_i} g$ Lie--Poisson commute on $\mathfrak{g}$.

Theorems & Definitions (29)

  • Lemma 1: B1TrTh
  • Remark 1
  • Remark 2
  • Theorem 2
  • proof
  • Lemma 3: BJ2
  • Corollary 4
  • Definition 1
  • Theorem 5
  • proof
  • ...and 19 more