Left Regular Functions from the Quaternions to the Split-Quaternions

Ilwoo Cho

doi:10.33401/fujma.1588231

Left Regular Functions from the Quaternions to the Split-Quaternions

Abstract

In this paper, we study the regularity of $\mathbb{R}$-differentiable functions on open connected subsets of the scaled hypercomplex numbers $\left\{ \mathbb{H}_{t}\right\} _{t\in[-1.1]}$, in terms of the well-defined differential operators $\left\{ \nabla_{t}\right\} _{t\in\left[-1,1\right]}$, where $\left[-1,1\right]$ is the closed interval in $\mathbb{R}$. To do that, we concentrate on studying the kernels $\left\{ \mathrm{ker}\nabla_{t}\right\} _{t\in\left[-1,1\right]}$ of $\left\{ \nabla_{t}\right\} _{t\in[-1,1]}$. And then, we define and study scale-depending differential operator $\nabla_{\left[-1,1\right]}$ acting on the direct product $\mathbb{R}$-algebra $\mathscr{H}\left[-1,1\right]=\underset{t\in\left[-1,1\right]}{\oplus^{a}}\mathbb{H}_{t}$, and the corresponding kernel $\mathrm{ker}\nabla_{\left[-1,1\right]}$, where $\oplus^{a}$ means the pure-algebraic direct product of $\mathbb{R}$-algebras.

Keywords

References

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Details

Primary Language

English

Subjects

Operator Algebras and Functional Analysis

Journal Section

Research Article

Authors

Ilwoo Cho ^*
0000-0001-8962-1089
United States

Publication Date

September 30, 2025

Submission Date

November 19, 2024

Acceptance Date

June 26, 2025

Published in Issue

Year 2025 Volume: 8 Number: 3

DOI

https://doi.org/10.33401/fujma.1588231

IZ

https://izlik.org/JA93ZD38NE

Cite

RIS / Bibtex

APA

Cho, I. (2025). Left Regular Functions from the Quaternions to the Split-Quaternions. Fundamental Journal of Mathematics and Applications, 8(3), 115-147. https://doi.org/10.33401/fujma.1588231

AMA

1.Cho I. Left Regular Functions from the Quaternions to the Split-Quaternions. Fundam. J. Math. Appl. 2025;8(3):115-147. doi:10.33401/fujma.1588231

Chicago

Cho, Ilwoo. 2025. “Left Regular Functions from the Quaternions to the Split-Quaternions”. Fundamental Journal of Mathematics and Applications 8 (3): 115-47. https://doi.org/10.33401/fujma.1588231.

EndNote

Cho I (September 1, 2025) Left Regular Functions from the Quaternions to the Split-Quaternions. Fundamental Journal of Mathematics and Applications 8 3 115–147.

IEEE

[1]I. Cho, “Left Regular Functions from the Quaternions to the Split-Quaternions”, Fundam. J. Math. Appl., vol. 8, no. 3, pp. 115–147, Sept. 2025, doi: 10.33401/fujma.1588231.

ISNAD

Cho, Ilwoo. “Left Regular Functions from the Quaternions to the Split-Quaternions”. Fundamental Journal of Mathematics and Applications 8/3 (September 1, 2025): 115-147. https://doi.org/10.33401/fujma.1588231.

JAMA

1.Cho I. Left Regular Functions from the Quaternions to the Split-Quaternions. Fundam. J. Math. Appl. 2025;8:115–147.

MLA

Cho, Ilwoo. “Left Regular Functions from the Quaternions to the Split-Quaternions”. Fundamental Journal of Mathematics and Applications, vol. 8, no. 3, Sept. 2025, pp. 115-47, doi:10.33401/fujma.1588231.

Vancouver

1.Ilwoo Cho. Left Regular Functions from the Quaternions to the Split-Quaternions. Fundam. J. Math. Appl. 2025 Sep. 1;8(3):115-47. doi:10.33401/fujma.1588231