Global Theory And Several Variables Theory Of Slice Hypercomplex Analysis

This dissertation mainly studies the global theory of slice hypercomplex analysis of several variables.Slice quaternionic analysis was introduced by Gentili and Struppa in 2006.This theory has been developed rapidly.Many classical results in complex analysis have been extended to slice quaternionic analysis.This theory also gives rise to the S-spectrum theory,which has important applications in quaternary quantum physics.However,the theory still lacks the global theory and slice sedenionic analysis.This is exactly the research content of this article.The global theory of slice hypercomplex analysis originates from complex analy-sis.It is a generalization of Riemann surfaces to the field of slice quaternionic analysis.The early research of this theory was greatly hindered due to any early erroneous re-search result.The erroneous result claims that any slice regular function defined on an s-domain in H can extend to be a slice regular function on a larger axially symmetric s-domain.This article gives a counterexample to this conclusion,which opens up a new gate to study the global theory of slice quaternionic analysis.Our results show that a slice regular function defined on an s-domain may have no its domain of existence in H.More importantly,we find that the topology of slice hypercomplex analysis is no longer Euclidean topology.The analogy of Riemann surfaces gives rise to a specific general-ized orbifold theory.In the study of the global theory of slice hypercomplex analysis,we have established a very important representation formula.This formula is different from the classical situation in that it depends on the path of continuation.Our research is limited to the generalization of Riemann domains in slice quaternion analysis.The theory of Riemann surfaces in the slice situation needs further research.There are very few results in the research of slice hypercomplex analysis in several variables.In the study of this theory,we propose a new method,which is to replace imaginary units in the algebra in consideration with complex structures.This method allows us to extend the existing slice analysis of real alternative*-algebra to sedenions and even Cayley-Dickson algebras.In the research of slice hypercomplex analysis in several variables,we adopt a two-step strategy so that we can attribute the problem to several complex analysis and the theory of representation formulas.Our method produces a new theory,called weak slice hypercomplex analysis.This theory is different from the existing strong slice hypercomplex analysis which relies on stem functions.To establish the theory of representation formulas,we have encountered great technical difficulties.That is the difference between two distinct imaginary units in the algebra in consideration may be not invertible.We solve this problem based on the theory of Moore-Penrose inverse.Our slice hypercomplex analysis in several variables is a generalization of several complex analysis in high-dimensional and non-commutative and even non-associate fields.That enriches the classical slice quaternionic analysis.It is worth emphasizing that the natural topology of this theory is no longer Euclidean topology,although it is a complex manifold on each slice.This theory inspires us to extend the theory of domains of holomorphy in several complex variables,and establish the corresponding theory in the slice hypercomplex analysis.This theory also inspired us to extend the theory of the Dolbeault complex to the slice hypercomplex analysis.