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An easier way to compute 2-cocycles coming from a reduction for semidirect products

Viacheslav Goncharov

TL;DR

The paper develops a general framework for computing central extensions in Hamiltonian actions of semidirect products $G = F ⋉ H$ by restricting to stabilisers under a key condition $\mathfrak{g}_{(\eta, a)} \oplus \mathfrak{h} = \mathfrak{g}$. It shows that the residual cocycle $c_{\rm res}$ on $\mathfrak{g}/\mathfrak{h}$ coincides with the stabiliser cocycle $c_{\rm stab}$ from restricting the action to $\mathfrak{g}_{(\eta, a)}$, provided the momentum maps agree on the relevant level set. This simplification is applied to infinite-dimensional settings, notably Teichmüller-type spaces arising from loop groups and circle diffeomorphisms, where the cocycle reduces to the Gelfand–Fuchs form, while some components may yield vanishing cocycles. The paper details three representative examples—coadjoint orbits, character varieties, and Teichmüller spaces—illustrating how the method recovers or clarifies the associated cocycles and connecting to Drinfeld–Sokolov-like reductions. The results provide a practical route to identify central extensions in complex Hamiltonian reduction problems with wide applicability in geometric representation theory and Teichmüller theory.

Abstract

For Hamiltonian actions of semidirect products $G=F \ltimes H$, we study 2-cocycles arising from residual Hamiltonian actions of $F$ on Hamiltonian reductions for $H$. The motivation comes from the study of Teichmuller spaces for surfaces with boundary, which carry Hamiltonian actions of the Virasoro algebra. In this paper, we give a general setup for the problem, and we suggest an easier way to obtain the Gelfand-Fuchs 2-cocycles for Hamiltonian actions on Teichmuller spaces.

An easier way to compute 2-cocycles coming from a reduction for semidirect products

TL;DR

The paper develops a general framework for computing central extensions in Hamiltonian actions of semidirect products by restricting to stabilisers under a key condition . It shows that the residual cocycle on coincides with the stabiliser cocycle from restricting the action to , provided the momentum maps agree on the relevant level set. This simplification is applied to infinite-dimensional settings, notably Teichmüller-type spaces arising from loop groups and circle diffeomorphisms, where the cocycle reduces to the Gelfand–Fuchs form, while some components may yield vanishing cocycles. The paper details three representative examples—coadjoint orbits, character varieties, and Teichmüller spaces—illustrating how the method recovers or clarifies the associated cocycles and connecting to Drinfeld–Sokolov-like reductions. The results provide a practical route to identify central extensions in complex Hamiltonian reduction problems with wide applicability in geometric representation theory and Teichmüller theory.

Abstract

For Hamiltonian actions of semidirect products , we study 2-cocycles arising from residual Hamiltonian actions of on Hamiltonian reductions for . The motivation comes from the study of Teichmuller spaces for surfaces with boundary, which carry Hamiltonian actions of the Virasoro algebra. In this paper, we give a general setup for the problem, and we suggest an easier way to obtain the Gelfand-Fuchs 2-cocycles for Hamiltonian actions on Teichmuller spaces.

Paper Structure

This paper contains 23 sections, 4 theorems, 55 equations.

Table of Contents

  1. Intoduction
  2. Acknowledgments.
  3. Introducing the problem
  4. Introductory definitions and the setup
  5. Description of the problem
  6. Special condition
  7. Intermediate results
  8. Main theorem
  9. Examples
  10. The group and the algebra.
  11. The functional.
  12. Remark.
  13. The first example: coadjoint orbit
  14. Loop group
  15. Diffeomorphisms of a circle
  16. ...and 8 more sections

Key Result

Lemma 1

Suppose that the groups $G, H, F$ and $G_{(\eta, a)}$ are connected and simply connected, and the functional $\eta\in\mathfrak{g}^*$ satisfies (TheCond). Then $G = G_{(\eta, a)}H = HG_{(\eta, a)}$.

Theorems & Definitions (8)

  • Lemma 1
  • proof
  • Proposition 1
  • proof
  • Theorem 1
  • proof
  • Proposition 2
  • proof