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On the Fueter-Sce theorem and Cauchy-Kovalevskaya extensions over alternative $\ast$-algebras

Qinghai Huo, Irene Sabadini, Zhenghua Xu

Abstract

Recently, the concept of generalized partial-slice monogenic (or regular) functions has been introduced and studied over Clifford algebras and octonions, respectively. In this paper, we further develop the theory of generalized partial-slice monogenic functions defined on hypercomplex subspaces and with values in a real alternative $\ast$-algebra and we concentrate on the Fueter-Sce theorem, three types of Cauchy-Kovalevskaya extensions, and their various internal relationships. The paper proposes more bridges between the theories of monogenicity, harmonicity, and generalized partial-slice monogenicity.

On the Fueter-Sce theorem and Cauchy-Kovalevskaya extensions over alternative $\ast$-algebras

Abstract

Recently, the concept of generalized partial-slice monogenic (or regular) functions has been introduced and studied over Clifford algebras and octonions, respectively. In this paper, we further develop the theory of generalized partial-slice monogenic functions defined on hypercomplex subspaces and with values in a real alternative -algebra and we concentrate on the Fueter-Sce theorem, three types of Cauchy-Kovalevskaya extensions, and their various internal relationships. The paper proposes more bridges between the theories of monogenicity, harmonicity, and generalized partial-slice monogenicity.
Paper Structure (15 sections, 31 theorems, 228 equations)

This paper contains 15 sections, 31 theorems, 228 equations.

Table of Contents

  1. Introduction
  2. Preliminary results
  3. Real alternative $\ast$-algebras
  4. Hypercomplex subspaces
  5. Generalized partial-slice monogenic functions
  6. Identity theorem and Representation Formula
  7. Generalized partial-slice functions
  8. Fueter polynomials
  9. CK-extensions
  10. CK-extension for generalized partial-slice monogenic functions
  11. Generalized CK-extension for monogenic functions
  12. Generalized CK-extension for harmonic functions
  13. Fueter-Sce theorem
  14. Connections between the Fueter-Sce theorem and CK-extensions
  15. Further remarks and future directions of research

Key Result

Proposition 2.1

(Schafer) For any $x, y \in \mathbb{A},$ if x is invertible, then it holds that

Theorems & Definitions (81)

  • Proposition 2.1
  • Proposition 2.2
  • Example 2.3: Division algebras
  • Example 2.4: Clifford algebras
  • Definition 2.5
  • Example 2.6
  • Proposition 2.7
  • Lemma 2.8
  • Definition 2.9
  • Example 2.10
  • ...and 71 more