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Reduction theory for stably graded Lie algebras

Jack A. Thorne

TL;DR

This work establishes a two-step reduction framework for representations arising from stable $Z/mZ$-gradings of semisimple Lie algebras. It defines a reduction covariant ${ rak R}:V^s(R) o X_G$ by associating to each stable vector a Cartan involution compatible with the grading, and shows how this covariant enables a notion of reducedness in the associated symmetric space. The authors then develop an explicit LLL-type reduction algorithm for $G= ext{SO}(2g+1)$ to move elements into a Siegel set, including a finite-termination proof via Weyl reflections; this is complemented by an explicit example in the stable $Z/2Z$-grading of ${ rak{sl}}_{2g+1}$ and a self-adjoint operator reduction illustrating the interplay with arithmetic applications such as 2-descent on Jacobians of odd hyperelliptic curves. Together, these results link representation-theoretic reduction to arithmetic invariants, enabling effective computation of integral orbits and connections to Selmer groups and the Jacobians of families of curves.

Abstract

We define a reduction covariant for the representations a la Vinberg associated to stably graded Lie algebras. We then give an analogue of the LLL algorithm for the odd split special orthogonal group and show how this can be combined with our theory to effectively reduce the coefficients of vectors in a representation connected to 2-descent for odd hyperelliptic curves.

Reduction theory for stably graded Lie algebras

TL;DR

This work establishes a two-step reduction framework for representations arising from stable -gradings of semisimple Lie algebras. It defines a reduction covariant by associating to each stable vector a Cartan involution compatible with the grading, and shows how this covariant enables a notion of reducedness in the associated symmetric space. The authors then develop an explicit LLL-type reduction algorithm for to move elements into a Siegel set, including a finite-termination proof via Weyl reflections; this is complemented by an explicit example in the stable -grading of and a self-adjoint operator reduction illustrating the interplay with arithmetic applications such as 2-descent on Jacobians of odd hyperelliptic curves. Together, these results link representation-theoretic reduction to arithmetic invariants, enabling effective computation of integral orbits and connections to Selmer groups and the Jacobians of families of curves.

Abstract

We define a reduction covariant for the representations a la Vinberg associated to stably graded Lie algebras. We then give an analogue of the LLL algorithm for the odd split special orthogonal group and show how this can be combined with our theory to effectively reduce the coefficients of vectors in a representation connected to 2-descent for odd hyperelliptic curves.
Paper Structure (9 sections, 10 theorems, 28 equations)

This paper contains 9 sections, 10 theorems, 28 equations.

Table of Contents

  1. Introduction
  2. Outline
  3. Notation
  4. The reduction covariant
  5. Background on Cartan involutions
  6. Application to graded Lie algebras
  7. Example: The stable $\mathbf{Z} / 2 \mathbf{Z}$-grading of ${\mathfrak{s}}{\mathfrak{l}}_{2g+1}$
  8. A reduction algorithm for $\operatorname{SO}(J)$
  9. Example: Reducing a self-adjoint linear operator

Key Result

Proposition 2.4

Let $H$ be a semisimple group over $\mathbf{R}$, and let $\theta$ be an involution of $H$. Then $\theta : H \to H$ is a Cartan involution if and only if $\theta : {\mathfrak h} \to {\mathfrak h}$ is a Cartan involution.

Theorems & Definitions (24)

  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • proof
  • Theorem 2.7
  • proof
  • Proposition 2.8
  • proof
  • Proposition 3.1
  • ...and 14 more