A Classification of Fueter-Regular Quaternionic Functions Based on the Complex Linearity of Their Real Differentials

Core Concepts

This paper classifies Fueter-regular quaternionic functions based on the degree of complex linearity exhibited by their real differentials when interacting with various almost-complex structures.

Abstract

• Bibliographic Information: Perotti, A., & Stoppato, C. (2024). Which Fueter-regular functions are holomorphic? arXiv preprint arXiv:2411.00127v1.
• Research Objective: This paper aims to classify Fueter-regular quaternionic functions by analyzing the complex linearity of their real differentials with respect to different almost-complex structures.
• Methodology: The authors utilize concepts from quaternionic analysis, linear algebra, and differential geometry. They leverage previous work on biregular functions and the notion of quaternionic linearity to develop a classification scheme based on the "size" of the real differential, which quantifies its complex linearity.
• Key Findings: The paper demonstrates that most Fueter-regular functions are not holomorphic with respect to any almost-complex structure. In cases where holomorphy exists, there is typically only one such structure. Functions holomorphic with respect to multiple structures are limited to conformal real affine transformations or constants. The authors provide a complete classification of regular real linear maps in terms of their size and complex linearity.
• Main Conclusions: This research provides a deeper understanding of the relationship between Fueter-regularity in quaternionic analysis and holomorphy in complex analysis. The classification scheme based on the size of the real differential offers a new perspective on the behavior of these functions.
• Significance: This work contributes significantly to the field of quaternionic analysis by providing a novel classification of Fueter-regular functions and clarifying their connection to holomorphic functions.
• Limitations and Future Research: The paper focuses on functions of one quaternionic variable. Exploring similar classifications for functions of several quaternionic variables could be a potential avenue for future research.

Stats

Quotes

Key Insights Distilled From

Which Fueter-regular functions are holomorphic?

by Alessandro P... at arxiv.org 11-04-2024

https://arxiv.org/pdf/2411.00127.pdf

Deeper Inquiries

How does this classification of Fueter-regular functions extend to the study of functions in several quaternionic variables?

Answer:
Extending the classification of Fueter-regular functions based on the complex linearity of their differentials to several quaternionic variables presents significant challenges. Here's a breakdown of the complexities and potential approaches:
Challenges:

Increased Dimensionality: With multiple quaternionic variables, the dimension of the domain and the tangent spaces increases considerably. This leads to a much richer structure of almost-complex structures, making a direct generalization of the single-variable classification quite intricate.
Interplay Between Variables: The interaction between different quaternionic variables introduces new complexities. The notion of complex linearity of the differential needs to be carefully adapted to account for these interactions.
Lack of a Canonical Choice: Unlike the complex case (C^n), there's no canonical choice of almost-complex structure in higher-dimensional quaternionic spaces (H^n). This makes it difficult to define a universally applicable notion of holomorphicity.
Potential Approaches:

Restricted Classes: One approach could be to focus on specific subclasses of Fueter-regular functions in several variables. For instance, one might consider functions that exhibit certain symmetries or satisfy additional constraints, simplifying the analysis of their differentials.
Generalized Complex Structures: Exploring generalized complex structures, which encompass both complex and symplectic structures, might offer a more suitable framework for studying holomorphicity in quaternionic spaces.
Algebraic Techniques: Employing tools from algebraic geometry and representation theory could provide insights into the structure of Fueter-regular functions and their differentials in higher dimensions.
Further Research:
This area is still under active research, and a complete classification of Fueter-regular functions in several quaternionic variables remains an open problem. Further investigation is needed to develop appropriate tools and techniques to tackle this challenge.

Could there be alternative classifications of Fueter-regular functions based on different properties other than the complex linearity of their differentials?

Answer:
Yes, alternative classifications of Fueter-regular functions are certainly possible by considering properties beyond the complex linearity of their differentials. Here are a few potential avenues:
1. Growth and Value Distribution:

Order and Type: Analogous to complex analysis, one could classify Fueter-regular functions based on their order and type, which characterize their growth at infinity.
Value Distribution Theory: Studying the distribution of values attained by Fueter-regular functions, similar to Nevanlinna theory in complex analysis, could lead to a classification based on their value distribution properties.
2. Geometric Properties:

Conformal Properties: While all Fueter-regular functions are harmonic, not all are conformal. Classifying them based on their conformal behavior (e.g., preserving angles, ratios of lengths) could be insightful.
Harmonic Morphisms: Investigating Fueter-regular functions that are also harmonic morphisms, meaning they pull back harmonic functions to harmonic functions, could provide a geometrically meaningful classification.
3. Algebraic and Representation-Theoretic Properties:

Symmetry Groups: Analyzing the symmetry groups of Fueter-regular functions, i.e., the groups of transformations that preserve their regularity, could lead to a classification based on their symmetry properties.
Representations: Viewing Fueter-regular functions as elements of certain function spaces, one could explore their properties under representations of relevant groups, potentially leading to a classification based on their representation-theoretic behavior.
4. Relationship to Other Function Theories:

Monogenic Functions: Exploring connections between Fueter-regular functions and monogenic functions in Clifford analysis could offer new perspectives and classification criteria.
Hypercomplex Analysis: Investigating Fueter-regular functions within the broader framework of hypercomplex analysis, which studies functions of hypercomplex variables, might reveal alternative classifications based on their properties in this generalized setting.
Exploring these alternative classifications could provide a deeper understanding of the structure and properties of Fueter-regular functions, potentially leading to new applications in various areas of mathematics and physics.

What implications does this connection between quaternionic analysis and complex analysis have for other areas of mathematics or physics?

Answer:
The connection between quaternionic analysis and complex analysis, as highlighted by the classification of Fueter-regular functions based on complex linearity, has profound implications for various areas of mathematics and physics. Here are some key examples:
Mathematics:

Differential Geometry: This connection provides new tools for studying the geometry of 4-manifolds. Fueter-regular functions can be used to construct special geometric structures, such as hyperkähler structures, which are important in string theory and mathematical physics.
Harmonic Analysis: The theory of Fueter-regular functions offers a rich extension of harmonic analysis to the quaternionic setting. It provides new insights into the properties of harmonic functions in four dimensions and their applications in areas like potential theory.
Clifford Analysis: The interplay between quaternionic and complex analysis strengthens the connections with Clifford analysis, a generalization of complex analysis to higher dimensions. This leads to a more unified approach to studying function theories in various settings.
Representation Theory: The classification of Fueter-regular functions based on complex linearity has implications for the representation theory of certain groups, such as the conformal group. It provides new insights into the structure of these groups and their representations.
Physics:

Quantum Mechanics: Quaternions have long been used to describe rotations and orientations in three dimensions. The connection with complex analysis suggests potential applications in quantum mechanics, particularly in the study of spin and angular momentum.
Field Theory: Fueter-regular functions and their generalizations have found applications in field theory, particularly in gauge theories and supersymmetry. They provide a natural framework for describing certain types of fields and their interactions.
Fluid Dynamics: The theory of quaternionic analysis, particularly the study of Fueter-regular functions, has applications in fluid dynamics, particularly in the study of ideal fluids and vortex dynamics.
Electromagnetism: Quaternions can be used to express Maxwell's equations in a compact and elegant form. The connection with complex analysis might lead to new insights into electromagnetic theory and its applications.
Overall, the interplay between quaternionic and complex analysis is a fertile ground for research, with the potential to advance our understanding of fundamental mathematical structures and physical phenomena.