Algebraic Function Theory in Quaternionic Spaces

Algebraic function theory in quaternionic spaces extends classical complex analysis into a noncommutative setting, offering a framework for functions of one or more quaternionic variables. Quaternions generalise complex numbers by introducing three distinct imaginary units whose multiplication does not commute, thereby necessitating novel definitions of analyticity and regularity. Central to this field are slice regular functions, defined so as to admit Cauchy‐type integral formulae and power‐series expansions on complex “slices” of the quaternionic space. Complementary to slice regularity are monogenic functions, which lie in the kernel of a Dirac‐type operator and satisfy a quaternionic analogue of the Cauchy–Riemann equations. The Fueter mapping theorem provides a systematic procedure to pass between slice regular and monogenic function spaces, and underpins the construction of spectral theories for noncommuting operators. In particular, the S‐functional calculus and its variants allow one to apply algebraic functions to quaternionic operators, with applications ranging from quaternionic quantum mechanics to signal processing, fractional diffusion, and the study of boundary‐value problems. The interplay between algebraic structures and functional‐analytic techniques has led to new insights into zero distributions, integral representations, and the geometry of function spaces in four and higher dimensions. This theory thus bridges pure algebraic concepts with concrete applications in physics and engineering, illustrating both its depth and its practical significance.

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Algebraic Function Theory in Quaternionic Spaces publication trend

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Technical terms

Quaternionic space: A four-dimensional real algebra generated by one real unit and three imaginary units whose multiplication is noncommutative.

Slice regular (hyperholomorphic) function: A function defined on quaternionic domains that admits a generalised Cauchy formula and power-series expansion on complex slices determined by the quaternionic imaginary units.

Monogenic function: A function taking values in the quaternionic algebra that satisfies a Dirac-type equation and serves as a hyperholomorphic analogue of holomorphic functions.

S-spectrum: The set of quaternionic values for which a noncommuting operator fails to be invertible under the S-functional calculus, generalising the usual operator spectrum.

Functional calculus: A method for assigning algebraic or analytic functions to operators, enabling the extension of polynomial and power-series operations to noncommuting contexts.

References

1. Slice regular functions in several variables. Mathematische Zeitschrift (2022).
2. Axially Harmonic Functions and the Harmonic Functional Calculus on the S-spectrum. The Journal of Geometric Analysis (2022).
3. The Fine Structure of the Spectral Theory on the S-Spectrum in Dimension Five. The Journal of Geometric Analysis (2023).
4. Slice monogenic functions of a Clifford variable via the S S -functional calculus. Proceedings of the American Mathematical Society Series B (2021).
5. An Introduction to Hyperholomorphic Spectral Theories and Fractional Powers of Vector Operators. Advances in Applied Clifford Algebras (2021).
6. On the Quaternionic Short-Time Fourier and Segal–Bargmann Transforms. Mediterranean Journal of Mathematics (2021).
7. The Noncommutative Fractional Fourier Law in Bounded and Unbounded Domains. Complex Analysis and Operator Theory (2021).
8. Riemann-Hilbert problems for monogenic functions in axially symmetric domains. Boundary Value Problems (2016).