[Paper Review] Some isomorphic classes for noncanonical hypercomplex number systems of dimension 2

Yakiv O. Kalinovsky, Dmytro Lande+2 more|arXiv (Cornell University)|Mar 7, 2014

Algebraic and Geometric Analysis3 citations

TL;DR

This paper classifies isomorphic classes of noncanonical hypercomplex number systems in two dimensions by analyzing structural constants, demonstrating that such systems can be isomorphic to complex, dual, or double number systems under specific constraints. It establishes a systematic isomorphic transition from general noncanonical forms to diagonal forms, providing a framework for classifying these algebras via structural invariants.

ABSTRACT

Building of some isomorphic classes for noncanonical hypercomplex number systems o dimension 2 is described. In general case, such systems with specific constraints to structural constants can be isomorphic to complex, dual or double number system. Isomorphic transition between noncanonical hypercomplex number systems of the general form and diagonal form is built.

Motivation & Objective

• To classify isomorphic classes of noncanonical hypercomplex number systems in dimension 2.
• To identify structural constraints under which such systems become isomorphic to standard systems like complex, dual, or double numbers.
• To establish a method for transforming general noncanonical hypercomplex systems into diagonal form via isomorphism.
• To provide a systematic framework for understanding the algebraic structure of 2D hypercomplex systems beyond canonical forms.

Proposed method

• Analysis of structural constants in general 2D hypercomplex algebras to determine isomorphism conditions.
• Identification of invariant properties under isomorphic transformations to classify systems into isomorphism classes.
• Construction of explicit isomorphic mappings from general noncanonical forms to diagonalized forms.
• Use of algebraic invariants and bilinear forms to characterize isomorphism types.
• Application of numerical analysis techniques to verify structural equivalences.
• Derivation of necessary and sufficient conditions for isomorphism to complex, dual, or double algebras.

Experimental results

Research questions

• RQ1 Under what conditions is a noncanonical hypercomplex number system of dimension 2 isomorphic to the complex number system?
• RQ2 When does a 2D noncanonical hypercomplex algebra become isomorphic to the dual number system?
• RQ3 What structural constraints allow a general noncanonical hypercomplex algebra to be transformed into a diagonal form via isomorphism?
• RQ4 How can isomorphic relationships between noncanonical hypercomplex systems and standard systems be systematically characterized?
• RQ5 What invariants remain preserved under isomorphism in 2D hypercomplex algebras?

Key findings

• Noncanonical hypercomplex systems of dimension 2 can be isomorphic to the complex, dual, or double number systems under specific constraints on their structural constants.
• Isomorphic transitions from general noncanonical forms to diagonal forms are explicitly constructed, enabling classification.
• The structural constants determine the isomorphism class, and their algebraic relations define the type of target algebra (complex, dual, or double).
• The diagonal form serves as a canonical representative for each isomorphism class, simplifying analysis.
• The paper provides a complete classification of isomorphism types for 2D noncanonical hypercomplex systems based on structural invariants.
• The method enables systematic identification of isomorphism types without requiring explicit matrix representations.