Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Invariant Theory Beyond Reductivity: Hilbert and Schwarz Theorems for Discrete Lorentz and Cocompact Groups

Leandro Nery

TL;DR

This work extends invariant theory beyond reductive and compact groups by analyzing two non-classical regimes: discrete subgroups of the Lorentz group $O(n,1)$ acting on $\mathbb{R}^{n,1}$ and cocompact actions on $\mathbb{R}^n$. It shows that for discrete Lorentz groups the polynomial invariant ring is finitely generated (a Hilbert-Weyl-type property) but Schwarz’s theorem fails to control smooth invariants, and for cocompact actions the polynomial invariants collapse to constants while smooth invariants obey a Schwarz-type description via a quotient $M/\Gamma$ and a set of smooth generators. The results jointly yield a fourfold classification of invariant-theoretic regimes, clarifying when algebraic generation and smooth generation align or diverge in non-compact settings. Overall, the paper delineates precise boundaries for the applicability of Hilbert-Weyl and Schwarz-type theorems beyond the classical reductive framework and highlights the distinct roles of algebra, analysis, and geometry in non-compact invariant theory.

Abstract

Classical invariant theory provides a complete framework for reductive and compact Lie groups, but the non-reductive and non-compact cases are significantly more complex and less completely understood. This work investigates two distinct regimes: discrete subgroups of the Lorentz group $O(n,1)$ acting on $\mathbb{R}^{n,1}$, and cocompact actions of discrete groups on $\mathbb{R}^n$. We prove that for discrete Lorentz groups, the ring of polynomial invariants remains finitely generated, yet the classical Schwarz Theorem, stating that smooth invariants are generated by polynomial ones, fails. Conversely, for cocompact actions, the polynomial invariant ring collapses to constants, but a persistent structure of smooth invariants persists and is finitely generated over the smooth functions on the quotient. These results reveal a fourfold classification of invariant-theoretic regimes, extending the classical framework and delineating the boundaries of the Hilbert-Weyl and Schwarz theorems.

Invariant Theory Beyond Reductivity: Hilbert and Schwarz Theorems for Discrete Lorentz and Cocompact Groups

TL;DR

This work extends invariant theory beyond reductive and compact groups by analyzing two non-classical regimes: discrete subgroups of the Lorentz group acting on and cocompact actions on . It shows that for discrete Lorentz groups the polynomial invariant ring is finitely generated (a Hilbert-Weyl-type property) but Schwarz’s theorem fails to control smooth invariants, and for cocompact actions the polynomial invariants collapse to constants while smooth invariants obey a Schwarz-type description via a quotient and a set of smooth generators. The results jointly yield a fourfold classification of invariant-theoretic regimes, clarifying when algebraic generation and smooth generation align or diverge in non-compact settings. Overall, the paper delineates precise boundaries for the applicability of Hilbert-Weyl and Schwarz-type theorems beyond the classical reductive framework and highlights the distinct roles of algebra, analysis, and geometry in non-compact invariant theory.

Abstract

Classical invariant theory provides a complete framework for reductive and compact Lie groups, but the non-reductive and non-compact cases are significantly more complex and less completely understood. This work investigates two distinct regimes: discrete subgroups of the Lorentz group acting on , and cocompact actions of discrete groups on . We prove that for discrete Lorentz groups, the ring of polynomial invariants remains finitely generated, yet the classical Schwarz Theorem, stating that smooth invariants are generated by polynomial ones, fails. Conversely, for cocompact actions, the polynomial invariant ring collapses to constants, but a persistent structure of smooth invariants persists and is finitely generated over the smooth functions on the quotient. These results reveal a fourfold classification of invariant-theoretic regimes, extending the classical framework and delineating the boundaries of the Hilbert-Weyl and Schwarz theorems.
Paper Structure (13 sections, 10 theorems, 33 equations, 1 table)

This paper contains 13 sections, 10 theorems, 33 equations, 1 table.

Table of Contents

  1. Introduction
  2. Statement of Main Results
  3. The Lorentz Regime: Polynomial Generation, Smooth Failure
  4. The Cocompact Regime: Polynomial Collapse, Smooth Generation
  5. Background: The Lorentz group and its discrete subgroups
  6. Discrete Lorentz group acting on Minkowski space
  7. $\Gamma$ as a Discrete Subgroup of $O(1,1)$
  8. $\Gamma$ is a discrete subgroup of $O(n,1)$
  9. The Schwarz Theorem and the Non-Compactness Barrier
  10. The Cocompact Case and the Hilbert-Weyl Barrier
  11. The Algebraic Obstruction: Periodicity vs. Polynomial Growth
  12. A Schwarz-Type Theorem for Cocompact Actions
  13. Synthesis: The Fourfold Landscape of Invariant Theory

Key Result

Theorem 3.1

Let $\Gamma$ be a discrete subgroup of $O(1,1)$ generated by a hyperbolic rotation $\mathcal{H}_\beta$ with fixed $\beta \in \mathbb{R} \setminus \{0\}$. Then the invariant ring $\mathcal{P}(\Gamma)$ is finitely generated and

Theorems & Definitions (26)

  • Remark 2.1
  • Theorem 3.1
  • proof
  • Remark 3.2
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • proof
  • Theorem 3.5
  • proof
  • ...and 16 more