SO(9,1) Group and Examples of Analytic Functions

Adv. Appl. Clifford Algebras

2020

In this paper we revisit the concept of conformality in the sense of Gauss in the context of octonions and Clifford algebras. We extend a characterization of conformality in terms of a system of partial differential equations and differential forms using special orthonormal sets of continuous functions that have been used before in the particular quaternionic setting. The aim is to describe to which higher dimensional algebras this characterization can exactly be extended and under which circumstances. It turns out to be crucial that this characterization requires a domain of definition that lies in a subalgebra that has the norm composition property and that is either associative (Clifford algebra case) or at least alternative (octonionic case). The orthonormal frames are elements of the spin group Spin$$(n+1)$$ ( n + 1 ) . We round off by relating the nature of the orthonormal frames to the associated Möbius transformation which are related to SO(9, 1) in the octonionic case and to the Ahlfors–Vahlen group in the case of a Clifford algebra.

Section: Resultsmentioning

confidence: 94%

Section: Resultsmentioning

confidence: 99%

“…For convenience we recall from [5,10,13,20] that the octonions offer an elegant representation of the SO(9, 1)-Möbius transformations (which can be R. S. Kraußhar Adv. Appl.…”

Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Conformal Mappings Revisited in the Octonions and Clifford Algebras of Arbitrary Dimension

Adv. Appl. Clifford Algebras

2020

Recent and new results on octonionic Bergman and Szegö kernels

Math Methods in App Sciences

2021

Very recently one has started to study Bergman and Szegö kernels in the setting of octonionic monogenic functions. In particular, explicit formulas for the Bergman kernel for the octonionic unit ball and for the octonionic right half‐space as well as a formula for the Szegö kernel for the octonionic unit ball have been established. In this paper, we extend this line of investigation by developing explicit formulas for the Szegö kernel of strip domains of the form from which we derive by a limit argument considering d → ∞ the Szegö kernel of the octonionic right half‐space. Additionally, we set up formulas for the Bergman kernel of such strip domains and relate both kernels with each other. In fact, these kernel functions can be expressed in terms of onefold periodic octonionic monogenic generalizations of the cosecant function and the cotangent function, respectively.

Towards Discrete Octonionic Analysis

Springer Proceedings in Mathematics &Amp; Statistics

Legatiuk

2023

In this paper, we finish the basic development of the discrete octonionic analysis by presenting a Weyl calculus-based approach to bounded domains in R 8 . In particular, we explicitly prove the discrete Stokes formula for a bounded cuboid, and then we generalise this result to arbitrary bounded domains in interior and exterior settings by the help of characteristic functions. After that, discrete interior and exterior Borel-Pompeiu and Cauchy formulae are introduced. Finally, we recall the construction of discrete octonionic Hardy spaces for bounded domains. Moreover, we explicitly explain where the non-associativity of octonionic multiplication is essential and where it is not. Thus, this paper completes the basic framework of the discrete octonionic analysis introduced in previous papers, and, hence, provides a solid foundation for further studies in this field.