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Gelfand-Cetlin abelianizations of symplectic quotients

Peter Crooks, Jonathan Weitsman

TL;DR

The paper develops an abelianization framework for generic symplectic quotients by relating them to torus quotients via Gelfand--Cetlin data. Central to the approach is a GC datum $( ext{lambda}_{ ext{big}}, rak{g}^*_{ ext{s-reg}})$ that endows an open dense subset of $ rak{g}^*$ with a Poisson $ ext{T}_{ ext{big}}$-space structure and yields a canonical torus action on $oldsymbol{ extmu}^{-1}( rak{g}^*_{ ext{s-reg}})$. The main result shows that, for $oldsymbol{\xi} ext{ in } rak{g}^*_{ ext{s-reg}}$, the symplectic quotient $M///_{oldsymbol{\xi}}G$ is canonically isomorphic to the torus quotient $M_{ ext{s-reg}}///_{ ext{lambda}_{ ext{big}}(oldsymbol{\xi})} ext{T}_{ ext{big}}$ (as smooth or stratified spaces, depending on freeness). The work extends to stratified symplectic spaces, giving a broad abelianization theorem and connecting classical GC systems (e.g., for $ ext{U}(n)$ and $ ext{SO}(n)$) with general compact Lie groups, via Thimm-type constructions and the Hoffman–Lane generalizations. This framework provides a systematic way to study generic quotients through integrable systems on $ rak{g}^*$, with potential implications for concrete computations of quotients in classical and quantum settings.

Abstract

We show that generic symplectic quotients of a Hamiltonian $G$-space $M$ by the action of a compact connected Lie group $G$ are also symplectic quotients of the same manifold $M$ by a compact torus. The torus action in question arises from certain integrable systems on $\mathfrak{g}^*$, the dual of the Lie algebra of $G$. Examples of such integrable systems include the Gelfand-Cetlin systems of Guillemin-Sternberg in the case of unitary and special orthogonal groups, and certain integrable systems constructed for all compact connected Lie groups by Hoffman-Lane. Our abelianization result holds for smooth quotients, and more generally for quotients which are stratified symplectic spaces in the sense of Sjamaar-Lerman.

Gelfand-Cetlin abelianizations of symplectic quotients

TL;DR

The paper develops an abelianization framework for generic symplectic quotients by relating them to torus quotients via Gelfand--Cetlin data. Central to the approach is a GC datum that endows an open dense subset of with a Poisson -space structure and yields a canonical torus action on . The main result shows that, for , the symplectic quotient is canonically isomorphic to the torus quotient (as smooth or stratified spaces, depending on freeness). The work extends to stratified symplectic spaces, giving a broad abelianization theorem and connecting classical GC systems (e.g., for and ) with general compact Lie groups, via Thimm-type constructions and the Hoffman–Lane generalizations. This framework provides a systematic way to study generic quotients through integrable systems on , with potential implications for concrete computations of quotients in classical and quantum settings.

Abstract

We show that generic symplectic quotients of a Hamiltonian -space by the action of a compact connected Lie group are also symplectic quotients of the same manifold by a compact torus. The torus action in question arises from certain integrable systems on , the dual of the Lie algebra of . Examples of such integrable systems include the Gelfand-Cetlin systems of Guillemin-Sternberg in the case of unitary and special orthogonal groups, and certain integrable systems constructed for all compact connected Lie groups by Hoffman-Lane. Our abelianization result holds for smooth quotients, and more generally for quotients which are stratified symplectic spaces in the sense of Sjamaar-Lerman.
Paper Structure (20 sections, 12 theorems, 60 equations)

This paper contains 20 sections, 12 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Organization
  4. Background and conventions
  5. Tori
  6. General compact connected Lie groups
  7. Hamiltonian geometry
  8. Gelfand--Cetlin data
  9. Definition and relation to integrable systems
  10. Existence of Gelfand--Cetlin data
  11. Construction of Gelfand--Cetlin data: integrality
  12. Construction of Gelfand--Cetlin data: Thimm's method
  13. Example of Gelfand--Cetlin datum: the Gelfand-Cetlin system on $\mathfrak{u}(n)^*$
  14. The abelianization theorem
  15. The universal maximal torus
  16. ...and 5 more sections

Key Result

Theorem 1

Let $G$ be a compact connected Lie group, and $M$ a Hamiltonian $G$-space with moment map $\mu:M\longrightarrow\mathfrak{g}^*$. Suppose that $(\lambda_{\emph{big}},\mathfrak{g}^*_{\emph{s-reg}})$ is a Gelfand--Cetlin datum, and consider a point $\xi\in\mathfrak{g}^*_{\emph{s-reg}}$.

Theorems & Definitions (25)

  • Theorem
  • Definition 1
  • Remark 2
  • Proposition 3
  • proof
  • Proposition 4
  • proof
  • Proposition 5
  • Proposition 6
  • Proposition 7
  • ...and 15 more