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Group actions on $p$-adic symplectic manifolds

Luis Crespo, Álvaro Pelayo

TL;DR

This work develops the theory of group actions on $p$-adic symplectic manifolds, proving that every $p$-adic symplectic action is weakly Hamiltonian via a momentum map $\mu:M\to\mathfrak{g}^*$. It further shows that, for Abelian $G$ acting properly, Hamiltonian actions are precisely those with isotropic orbits, highlighting a sharp contrast with the real setting. The authors introduce and analyze the $p$-adic torus and $p$-adic symplectic toric manifolds, establishing existence results, weak isomorphism flexibility via $GL(k,\mathbb{Z}_p)$, and a range of fixed-point phenomena that reveal a rich classification landscape. A variety of examples illustrate both Hamiltonian and non-Hamiltonian behavior, including a non-proper action with isotropic orbits, underscoring the necessity of properness and the unique features of the $p$-adic context. Overall, the paper extends symplectic concepts to $p$-adic geometry, delineating fundamental differences from the real case and opening new directions for toric and integrable systems in the $p$-adic setting.

Abstract

Let $p$ be a prime number. We introduce symplectic actions of $p$-adic analytic Lie groups on $p$-adic symplectic manifolds. Then we show that any $p$-adic symplectic action $G\times(M,ω)\to(M,ω)$ has a momentum map $μ:M\to\mathfrak{g}^*$, and that a proper $p$-adic symplectic action is Hamiltonian if and only if every orbit is isotropic. We conclude by defining $p$-adic symplectic toric manifolds, by analogy with the real case.

Group actions on $p$-adic symplectic manifolds

TL;DR

This work develops the theory of group actions on -adic symplectic manifolds, proving that every -adic symplectic action is weakly Hamiltonian via a momentum map . It further shows that, for Abelian acting properly, Hamiltonian actions are precisely those with isotropic orbits, highlighting a sharp contrast with the real setting. The authors introduce and analyze the -adic torus and -adic symplectic toric manifolds, establishing existence results, weak isomorphism flexibility via , and a range of fixed-point phenomena that reveal a rich classification landscape. A variety of examples illustrate both Hamiltonian and non-Hamiltonian behavior, including a non-proper action with isotropic orbits, underscoring the necessity of properness and the unique features of the -adic context. Overall, the paper extends symplectic concepts to -adic geometry, delineating fundamental differences from the real case and opening new directions for toric and integrable systems in the -adic setting.

Abstract

Let be a prime number. We introduce symplectic actions of -adic analytic Lie groups on -adic symplectic manifolds. Then we show that any -adic symplectic action has a momentum map , and that a proper -adic symplectic action is Hamiltonian if and only if every orbit is isotropic. We conclude by defining -adic symplectic toric manifolds, by analogy with the real case.

Paper Structure

This paper contains 16 sections, 18 theorems, 94 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Symplectic, weakly Hamiltonian and Hamiltonian actions: definitions
  3. $p$-adic analytic Lie groups
  4. $p$-adic analytic Lie group actions: symplectic and (weakly) Hamiltonian
  5. $p$-adic momentum maps and proof of Theorem \ref{['thm:weak']}
  6. Properties of $p$-adic momentum maps
  7. Proof of Theorem \ref{['thm:weak']}
  8. Proof of Theorem \ref{['thm:hamiltonian']}: $p$-adic momentum maps and isotropic orbits
  9. The $p$-adic torus and $p$-adic symplectic toric manifolds
  10. The $p$-adic torus
  11. $p$-adic symplectic toric manifolds: definition
  12. Results on $p$-adic symplectic toric actions
  13. Examples of $p$-adic symplectic/Hamiltonian actions
  14. A symplectic non-Hamiltonian $p$-adic action with isotropic orbits and proof of Theorem \ref{['thm:non-hamiltonian']}
  15. Final remarks
  16. ...and 1 more sections

Key Result

Theorem A

Let $p$ be a prime number. Let $G$ be a $p$-adic analytic Lie group with Lie algebra $\mathfrak{g}$. Let $(M,\omega)$ be a paracompact $p$-adic analytic symplectic manifold and let $\psi:G\times M\to M$ be a $p$-adic analytic symplectic Lie group action on $(M,\omega)$. Then $\psi:G\times M\to M$ is

Figures (8)

  • Figure 1: A comparison between the properties of proper Abelian Lie group actions on compact real symplectic manifolds (left) and proper Abelian Lie group actions on compact $p$-adic analytic symplectic manifolds (right), see Theorems \ref{['thm:weak']} and \ref{['thm:hamiltonian']} and Examples \ref{['ex:oscillator']}, \ref{['ex:spin']} and \ref{['ex:translation2']} for the diagram on he right-hand side.
  • Figure 2: Step 1 of the proof of Theorem \ref{['thm:hamiltonian']}. The red circle around $\mathbf{1}\in G$ represents $H_i$. For $q\in M$, the horizontal line represents $G_q$, the stabilizer of $q$, and the orange band is $S_q=G_qH_i$. The blue line represents $S_qq$, which is the same as $H_iq$. We have marked a point $m$ near $q$; then the cylinder is $S_qm$ and the purple curve is $G_qm$ (when $m$ tends to $q$, the curve collapses to a point and the cylinder to the blue line).
  • Figure 3: Step 2 of the proof of Theorem \ref{['thm:hamiltonian']}. For each $k\in\mathbb{N}$, $(m_k,m_k')$ is a bad pair contained in $U_{q,k}$, $g_k$ is such that $\psi(g_k,m_k)=m_k'$, $g_0$ is their limit, and $g_k'=g_0^{-1}g_k$.
  • Figure 4: Step 4a of the proof of Theorem \ref{['thm:hamiltonian']}. The set $U_q$ is split into $U_q'$ and $U_q"$, and we are defining $\mu$ in $U_q'$ by copying its value from a previous set $U_{q'}$.
  • Figure 5: Step 4b of the proof of Theorem \ref{['thm:hamiltonian']}. The orbit of $m$ is contained in $U_q"$, and we need to prove that $\mu$ is constant along the purple curve $G_qm$ and in the brown set $H_im$.
  • ...and 3 more figures

Theorems & Definitions (56)

  • Theorem A: Every $p$-adic symplectic action has a momentum map
  • Theorem B: Characterization of $p$-adic Hamiltonian actions
  • Theorem C: Example of non-Hamiltonian $p$-adic symplectic action
  • Definition 2.1: $p$-adic analytic Lie group Schneider
  • Definition 2.2: $p$-adic symplectic Lie group action
  • Definition 2.3: $p$-adic vector field generated by an action
  • Definition 2.4: $p$-adic Hamiltonian Lie group action
  • Remark 2.5
  • Proposition 3.1
  • proof
  • ...and 46 more