Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The symplectic form associated to a singular Poisson algebra

Hans-Christian Herbig, William Osnayder Clavijo Esquivel, Christopher Seaton

TL;DR

The paper develops a framework to define and realize symplectic forms on singular affine Poisson varieties via the naive de Rham complex of Lie-Rinehart algebras, addressing the obstacle of non-projective derivation modules in the singular setting. It constructs a Hamiltonian 2-form $$\omega^{\mathrm{Ham}}$$ and a nondegenerate 2-form $$\omega$$ in concrete cases, starting with the double cone, and then provides invariant-theoretic and orbifold-based extensions to simple cones and linear symplectic orbifolds. The authors connect these constructions with orbit-space lifting and differential-space formalisms, showing that the invariant de Rham complex supports closed, nondegenerate symplectic structures on singular quotients. The work suggests a path toward a singular Marsden–Weinstein-type theory and broader applications to symplectic singularities and moduli spaces, while highlighting computational approaches (e.g., Gröbner bases) as practical tools for explicit derivation modules. Overall, it establishes a principled method to extend symplectic geometry into singular realms and opens directions for further structural and physical applications.

Abstract

Given an affine Poisson algebra, that is singular one may ask whether there is an associated symplectic form. In the smooth case the answer is obvious: for the symplectic form to exist the Poisson tensor has to be invertible. In the singular case, however, derivations do not form a projective module and the nondegeneracy condition is more subtle. For a symplectic singularity one may naively ask if there is indeed an analogue of a symplectic form. We examine an example of a symplectic singularity, namely the double cone, and show that here such a symplectic form exists. We use the naive de Rham complex of a Lie-Rinehart algebra. Our analysis of the double cone uses Gröbner bases calculations. We also give an alternative construction of the symplectic form that generalizes to categorical quotients of cotangent lifted representations of finite groups. We use the same formulas to construct a symplectic form on the simple cone, seen as a Poisson differential space and generalize the construction to linear symplectic orbifolds. We present useful auxiliary results that enable to explicitly determine generators for the module of derivations an affine variety. The latter may be understood as a differential space.

The symplectic form associated to a singular Poisson algebra

TL;DR

The paper develops a framework to define and realize symplectic forms on singular affine Poisson varieties via the naive de Rham complex of Lie-Rinehart algebras, addressing the obstacle of non-projective derivation modules in the singular setting. It constructs a Hamiltonian 2-form and a nondegenerate 2-form in concrete cases, starting with the double cone, and then provides invariant-theoretic and orbifold-based extensions to simple cones and linear symplectic orbifolds. The authors connect these constructions with orbit-space lifting and differential-space formalisms, showing that the invariant de Rham complex supports closed, nondegenerate symplectic structures on singular quotients. The work suggests a path toward a singular Marsden–Weinstein-type theory and broader applications to symplectic singularities and moduli spaces, while highlighting computational approaches (e.g., Gröbner bases) as practical tools for explicit derivation modules. Overall, it establishes a principled method to extend symplectic geometry into singular realms and opens directions for further structural and physical applications.

Abstract

Given an affine Poisson algebra, that is singular one may ask whether there is an associated symplectic form. In the smooth case the answer is obvious: for the symplectic form to exist the Poisson tensor has to be invertible. In the singular case, however, derivations do not form a projective module and the nondegeneracy condition is more subtle. For a symplectic singularity one may naively ask if there is indeed an analogue of a symplectic form. We examine an example of a symplectic singularity, namely the double cone, and show that here such a symplectic form exists. We use the naive de Rham complex of a Lie-Rinehart algebra. Our analysis of the double cone uses Gröbner bases calculations. We also give an alternative construction of the symplectic form that generalizes to categorical quotients of cotangent lifted representations of finite groups. We use the same formulas to construct a symplectic form on the simple cone, seen as a Poisson differential space and generalize the construction to linear symplectic orbifolds. We present useful auxiliary results that enable to explicitly determine generators for the module of derivations an affine variety. The latter may be understood as a differential space.
Paper Structure (8 sections, 13 theorems, 50 equations, 1 figure)

This paper contains 8 sections, 13 theorems, 50 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Lie-Rinehart algebras and the naive de Rham complex
  3. The main idea
  4. The double cone
  5. Explanation in terms of orbifold symplectic forms
  6. The simple cone as a Poisson differential space
  7. Linear symplectic orbifolds
  8. Conclusion and outlook

Key Result

Theorem 3.1

Simis If $I\subseteq P$ be an ideal in a polynomial $\boldsymbol{k}$-algebra $P$ then $\operatorname{Der}( A) \simeq \operatorname{Der}_{I}( P) /I\operatorname{Der}( P)$, where $\operatorname{Der}_{I}( P)=\{X\in \operatorname{Der}(P)| X(I)\subseteq I\}$.

Figures (1)

  • Figure 1: The double cone.

Theorems & Definitions (23)

  • Theorem 3.1
  • Lemma 3.2
  • proof
  • Definition 3.3
  • Proposition 3.4
  • proof
  • Proposition 4.1
  • proof
  • Proposition 5.1
  • proof
  • ...and 13 more