Octonion Analysis of Several Variables
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Octonion Analysis of Several Variables Haiyan Wang · Guangbin Ren
Received: 22 May 2014 / Accepted: 10 September 2014 / Published online: 11 October 2014 © School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag Berlin Heidelberg 2014
Abstract The octonions are distinguished in the M-theory in which Universe is the usual Minkowski space R4 times a G 2 manifold of very small diameter with G 2 being the automorphism group of the octonions. The multidimensional octonion analysis is initiated in this article, which extends the theory of several complex variables, such as the Bochner–Martinelli formula, the theory of non-homogeneous Cauchy–Riemann equations, and the Hartogs principle, to the non-commutative and non-associative realm. Keywords Several octonionic variables · Bochner–Martinelli formula · Hartogs theorem · Non-homogenous Cauchy–Riemann equations Mathematics Subject Classification
35G20 · 30G35 · 32G05
1 Introduction The importance of the octonions has been found in string theory, special theory of relativity, and quantum theory [3,4] since the birth of the octonions discovered in 1843 by Graves and constructed in 1845 by Cayley. It is known that the automorphism group of the octonion algebra is the exceptional simple Lie group G 2 , while the Mtheory claims that the model of Universe is the usual Minkowski space R4 times a G 2 manifold of very small diameter.
H. Wang School of Science, Tianjin University of Technology and Education, Tianjin 300222, China e-mail: [email protected] G. Ren (B) Department of Mathematics, University of Science and Technology of China, Hefei 230026, China e-mail: [email protected]
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H. Wang, G. Ren
The octonion analysis is quite subtle due to non-commutative and non-associative feature. It is only restricted to the setting of one octonionic variable and is in its primary stage [13]. Comparing to the recently well-developed theory of several quaternionic theory for Cauchy–Fueter operators [5,7,16], nearly nothing have been done about multidimensional octonion analysis. In this paper, we study the octonion analysis of several variables and extend the theory of several complex variables to the octonions. In particular, we shall construct the explicit form of the Bochner–Martinelli integral representation formula related to several Dirac operators, solve the system of non-homogeneous Cauchy–Riemann equations of the octonionic version, and establish the Hartogs theorem on removability of compact singularity for regular functions in the octonion analysis of several variables. The Bochner–Martinelli integral representation formula is the key in the integral theory of the octonionic analysis of several variables. We refer to [9,11,12] for the version of several variables and its applications. The solvability of the non-homogeneous Cauchy–Riemann equations will lay the foundation for constructing functions in the octonionic analysis of several variables. As it is well known that many of the differences between the one and sev
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