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Invariants and Automorphisms for slice regular functions

Cinzia Bisi, Joerg Winkelmann

TL;DR

This work analyzes invariants under automorphisms of the algebra of slice regular functions over the quaternions and the Clifford algebra R_3 (with R_3 ≅ H ⊕ H). By introducing and leveraging the central divisor cdiv, together with stem-function constructions, the authors establish when two slice-regular data sets are related by an automorphism of the complexified algebra Aut(A_C); this yields a precise, holomorphically governed criterion that extends beyond Tr and N invariants. The strategy hinges on reducing to traceless components, applying local-to-global holomorphic section arguments on Stein domains, and carefully treating isotropy in the automorphism groups. The results give a structured, cohomology-backed description of Aut of slice-regular function algebras and connect automorphism invariants to complexification phenomena and central-divisor data, with explicit treatment of the R_3 case via its two-H-summand decomposition.

Abstract

Let $A$ be one of the following Clifford algebras : $\mathbb{R}_2 \cong \mathbb{H}$ or $\mathbb{R}_3$. For the algebra $A$, the automorphism group $Aut(A)$ and its invariants are well known. In this paper we will describe the invariants of the automorphism group of the algebra of slice regular functions over $A$.

Invariants and Automorphisms for slice regular functions

TL;DR

This work analyzes invariants under automorphisms of the algebra of slice regular functions over the quaternions and the Clifford algebra R_3 (with R_3 ≅ H ⊕ H). By introducing and leveraging the central divisor cdiv, together with stem-function constructions, the authors establish when two slice-regular data sets are related by an automorphism of the complexified algebra Aut(A_C); this yields a precise, holomorphically governed criterion that extends beyond Tr and N invariants. The strategy hinges on reducing to traceless components, applying local-to-global holomorphic section arguments on Stein domains, and carefully treating isotropy in the automorphism groups. The results give a structured, cohomology-backed description of Aut of slice-regular function algebras and connect automorphism invariants to complexification phenomena and central-divisor data, with explicit treatment of the R_3 case via its two-H-summand decomposition.

Abstract

Let be one of the following Clifford algebras : or . For the algebra , the automorphism group and its invariants are well known. In this paper we will describe the invariants of the automorphism group of the algebra of slice regular functions over .

Paper Structure

This paper contains 36 sections, 25 theorems, 132 equations.

Table of Contents

  1. Introduction
  2. Related work
  3. Preparations
  4. Conjugation, norm and trace
  5. Relation with notions in linear algebra and number theory
  6. Stem functions
  7. Compatibility
  8. Other notions
  9. An example
  10. Another example
  11. More preparations
  12. The Clifford algebra ${\mathbb R}_3$
  13. Invariants for ${\mathbb R}_3$
  14. Slice preserving functions
  15. Slice regular functions for $A={\mathbb R}_3$
  16. ...and 21 more sections

Key Result

Theorem 1.1

Let ${\mathbb H}$ denote the algebra of quaternions, ${\mathbb H}_{{\mathbb C}}={\mathbb H}{\otimes}_{\mathbb R}{\mathbb C}$, $G=Aut({\mathbb H})\cong SO(3,{\mathbb R})$, $G_{{\mathbb C}}=Aut({\mathbb H}_{{\mathbb C}})\cong SO(3,{\mathbb C})$. Let $D\subset{\mathbb C}$ be a symmetric domain and let

Theorems & Definitions (55)

  • Theorem 1.1
  • Remark
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • Example 3.1
  • Proposition 5.1
  • Definition 5.2
  • proof
  • ...and 45 more