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Rational Curves on Real Classical Groups

Zijia Li, Ke Ye

TL;DR

This work develops a comprehensive theory of rational curves on real classical groups, delivering (i) a structure theorem that reduces quadratic rational curves on $G_B(\mathbb{F})$ to indecomposable blocks, (ii) a decomposition theorem that noncommutatively generalizes partial fraction decomposition, and (iii) generalized Kempe-type universality results for rational curves on homogeneous spaces. By classifying quadratic rational curves on unitary and (indefinite) orthogonal groups and extending to inhomogeneous groups, the authors provide explicit normal forms and block-structure parametrizations, with Sylvester-equation techniques underpinning the constructions. The generalized Kempe theorems establish that any rational curve on a homogeneous space can be realized as a product of low-degree rational curves on the acting group, and they extend to loop settings via lifting and approximation criteria. Overall, the paper lays foundational tools for understanding parametrizations of real Lie groups and their homogeneous spaces, with potential implications for kinematics, physics, and geometric modeling.

Abstract

This paper is concerned with rational curves on real classical groups. Our contributions are three-fold: (i) We determine the structure of quadratic rational curves on real classical groups. As a consequence, we completely classify quadratic rational curves on $\mathrm{U}_n$, $\mathrm{O}_n(\mathbb{R})$, $\mathrm{O}_{n-1,1}(\mathbb{R})$ and $\mathrm{O}_{n-2,2}(\mathbb{R})$. (ii) We prove a decomposition theorem for rational curves on real classical groups, which can be regarded as a non-commutative generalization of the fundamental theorem of algebra and partial fraction decomposition. (iii) As an application of (i) and (ii), we generalize Kempe's Universality Theorem to rational curves on homogeneous spaces.

Rational Curves on Real Classical Groups

TL;DR

This work develops a comprehensive theory of rational curves on real classical groups, delivering (i) a structure theorem that reduces quadratic rational curves on to indecomposable blocks, (ii) a decomposition theorem that noncommutatively generalizes partial fraction decomposition, and (iii) generalized Kempe-type universality results for rational curves on homogeneous spaces. By classifying quadratic rational curves on unitary and (indefinite) orthogonal groups and extending to inhomogeneous groups, the authors provide explicit normal forms and block-structure parametrizations, with Sylvester-equation techniques underpinning the constructions. The generalized Kempe theorems establish that any rational curve on a homogeneous space can be realized as a product of low-degree rational curves on the acting group, and they extend to loop settings via lifting and approximation criteria. Overall, the paper lays foundational tools for understanding parametrizations of real Lie groups and their homogeneous spaces, with potential implications for kinematics, physics, and geometric modeling.

Abstract

This paper is concerned with rational curves on real classical groups. Our contributions are three-fold: (i) We determine the structure of quadratic rational curves on real classical groups. As a consequence, we completely classify quadratic rational curves on , , and . (ii) We prove a decomposition theorem for rational curves on real classical groups, which can be regarded as a non-commutative generalization of the fundamental theorem of algebra and partial fraction decomposition. (iii) As an application of (i) and (ii), we generalize Kempe's Universality Theorem to rational curves on homogeneous spaces.
Paper Structure (18 sections, 40 theorems, 134 equations, 3 tables)

This paper contains 18 sections, 40 theorems, 134 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Rational curves on real algebraic varieties
  4. Quadratic rational curves on GB
  5. Structure theorem for quadratic rational curves
  6. Quadratic rational curves on unitary groups
  7. Quadratic rational curves on real orthogonal groups
  8. Quadratic rational curves on real indefinite orthogonal groups
  9. Decomposition of rational curves on linear algebraic groups
  10. Decomposition of rational curves on GB
  11. Decomposition of rational curves on IGB
  12. Generalizations of Kempe's Universality Theorem
  13. Generalized Kempe's Universality Theorem for loops
  14. Generalized Kempe's Universality Theorem
  15. Examples of small dimensions
  16. ...and 3 more sections

Key Result

Theorem 1

For any quadratic rational curve $\alpha$ on $G_B(\mathbb{F})$, there exist $R\in \mathop{\mathrm{GL}}\nolimits_n(\mathbb{F})$, $a\in \mathbb{R}$ and $b\in \mathbb{R} \setminus \{0\}$ such that $B = R \mathop{\mathrm{diag}}\nolimits(B_1,\dots, B_s) R^{\sigma}$ and where for each $1 \le p, q \le s$, $(X_p , B_p)$ is given in Table Tab:indecomposable and $Y_{pq}$ is given in Tables Tab:Candidates

Theorems & Definitions (87)

  • Theorem 1: \ref{['thm:structure']} (Structure theorem)
  • Theorem 2: \ref{['thm:KempeGBF']} (Decomposition theorem)
  • Theorem 3: \ref{['thm:Kempe G-space']} (Generalized Kempe's Universality Theorem II)
  • Lemma 2.1
  • Proposition 2.2
  • Theorem 2.3
  • Definition 3.1: rational curve
  • Lemma 3.2
  • Remark 3.3
  • Definition 3.4: degree
  • ...and 77 more