Left Regular Functions from the Quaternions to the Split-Quaternions

Year 2025, Volume: 8 Issue: 3, 115 - 147, 30.09.2025

Ilwoo Cho

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

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

Scaled Hypercomplexes , Differential Operators. , Regular functions

References

• [1] D. Alpay and I. Cho, Dynamical systems of operators induced by scaled hypercomplex rings, Adv. Appl. Clifford Algebras, 33(3) (2023), 33. $\href{https://doi.org/10.1007/s00006-023-01272-0}{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/85160110602?origin=resultslist}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000991977000002}{\mbox{[Web of Science]}} $
• [2] D. Alpay I.L. Paiva, and D.C. Struppa, A general setting for functions of Fueter variables: differentiability, rational functions, Fock module and related topics, Isr. J. Math., 236(1) (2020), 207-246. $ \href{https://doi.org/10.1007/s11856-020-1970-7}{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/85078318599?origin=resultslist}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000507795300002}{\mbox{[Web of Science]}} $
• [3] D. Alpay and M. Shapiro, Probleme de Gleason et interpolation pour les fonctions hyper-analytiques, C. R. Math., 335(11) (2002), 889-894. $ \href{https://doi.org/10.1016/S1631-073X(02)02598-0}{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/0038053069?origin=resultslist}{\mbox{[Scopus]}} $
• [4] M.L. Curtis, Matrix Groups, Springer-Verlag, New York, (1979), ISBN: 0-387-90462-X. $ \href{https://doi.org/10.1007/978-1-4612-5286-3}{\mbox{[CrossRef]}} $
• [5] R. Delanghe, On regular-analytic functions with values in a Clifford algebra, Math. Ann., 185 (1970), 91-111. $ \href{https://doi.org/10.1007/BF01359699}{\mbox{[CrossRef]}} $
• [6] V. Kravchenko, Applied Quaternionic Analysis, Heldemann Verlag, (2003), ISBN: 3-88538-228-8. $ \href{https://www.researchgate.net/publication/266226946_Applied_Quaternionic_Analysis}{\mbox{[Web]}} $
• [7] L. Rodman, Topics in Quaternion Linear Algebra, Princeton Univ. Press, NJ, (2014), ISBN: 978-0-691-16185-3. $ \href{https://doi.org/10.1515/9781400852741.fm}{\mbox{[CrossRef]}} $
• [8] A. Sudbery, Quaternionic Analysis, Math. Proc. Cambridge Philos. Soc., (1998). $ \href{https://doi.org/10.1017/S0305004100055638}{\mbox{[CrossRef]}} $
• [9] J. Voight, Quaternion Algebra, Dept. of Math., Dartmouth Univ., (2019). $ \href{https://doi.org/10.1007/978-3-030-56694-4}{\mbox{[CrossRef]}} $
• [10] D. Alpay M.E. Luna-Elizarraras, M. Shapiro and D. Struppa, Gleason’s problem, rational functions and spaces of left-regular functions: The split-quaternion setting, Isr. J. Math., 226(1) (2018), 319-349. $ \href{https://doi.org/10.1007/s11856-018-1696-y}{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/85046713003?origin=resultslist}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000437011800012}{\mbox{[Web of Science]}} $
• [11] I. Frenkel and M. Libine, Split quaternionic analysis and separation of the series for $SL(2, \mathbb{R})$ and $SL(2, \mathbb{C})/SL(2, \mathbb{R})$, Adv. Math., 228(2) (2011), 678-763. $ \href{https://doi.org/10.1016/j.aim.2011.06.001}{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/79960840421?origin=resultslist}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000293667800002}{\mbox{[Web of Science]}} $
• [12] M. Libine, An Invitation to Split Quaternionic Analysis, in Hypercomplex Analysis and Applications, Trends in Math., Birkh¨auser / Springer, Basel AG, Basel, (2011), 161–180. $ \href{https://doi.org/10.1007/978-3-0346-0246-4_12}{\mbox{[CrossRef]}} $
• [13] D. Alpay and I. Cho, Operators induced by certain hypercomplex systems, Opusc. Math., 43(3) (2023), 275-333. $ \href{https://doi.org/10.7494/OpMath.2023.43.3.275}{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/85164347227?origin=resultslist}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:001006037200001}{\mbox{[Web of Science]}} $
• [14] D. Alpay and I. Cho, On scaled hyperbolic numbers induced by scaled hypercomplex rings, Pure Appl. Funct. Anal., 9(6) (2024), 1397-1445. To Appear. $ \href{https://cris.bgu.ac.il/en/publications/on-scaled-hyperbolic-numbers-induced-by-scaled-hypercomplex-rings}{\mbox{[Web]}} $