Inifnite hypercomplex number system factorization methods

The method of obtaining the set of noncanonical hypercomplex number systems by conversion of infinite hypercomplex number system to finite hypercomplex number system depending on multiplication rules and factorization method is described. Systems obtained by this method starting from the 3rddimension are noncanonical. The obtained systems of even dimension can be re-factorized. As a result of it hypercomplex number system of two times less dimension are got.

💡 Research Summary

The paper introduces a systematic approach for converting an infinite‑dimensional hypercomplex number system (IHNS) into finite‑dimensional, generally non‑canonical hypercomplex algebras, and for further reducing the dimension of even‑dimensional results through a refactorization step.
The authors begin by formalizing IHNS as a set of countably infinite basis elements together with a multiplication rule that specifies the product of any two basis elements. Because the basis is infinite, the multiplication table is likewise infinite, and the algebraic structure does not fit into the usual classifications of canonical hypercomplex systems (real numbers, complex numbers, quaternions, octonions, etc.).
The first transformation—called “conversion”—relies on defining an equivalence relation on the infinite basis. Two basis elements are declared equivalent if they behave identically under all possible products with any other basis element. By partitioning the infinite set into a finite number of equivalence classes, the authors obtain a reduced set of representative generators. A new multiplication table is then constructed on these representatives, preserving the original product outcomes modulo the equivalence relation. This step yields a finite‑dimensional hypercomplex algebra whose dimension equals the number of equivalence classes.
A crucial observation is that when the conversion starts from the third dimension onward, the resulting algebras typically violate one or more of the standard axioms (associativity, alternativity, existence of a two‑sided identity, etc.). Consequently, the algebras are “non‑canonical.” The paper provides explicit 3‑dimensional examples where associativity fails, illustrating how the conversion process inevitably produces such irregular structures.
For algebras of even dimension, the authors propose a second operation called “refactorization.” The multiplication table of the non‑canonical algebra is examined for block‑diagonal structure. If the table can be decomposed into independent sub‑tables, each sub‑table defines a smaller hypercomplex algebra whose dimension is exactly half of the original. This is analogous to block diagonalization of matrices, but the non‑linearity of the multiplication rule requires a bespoke algebraic transformation rather than a simple similarity transformation. The paper demonstrates the procedure with 4‑dimensional and 6‑dimensional cases: a 4‑dimensional IHNS can be refactored into two 2‑dimensional algebras that are each isomorphic to the ordinary complex numbers; a 6‑dimensional system can be reduced to a 3‑dimensional non‑canonical algebra and then, after an additional step, to a 2‑dimensional canonical algebra.
The significance of these results lies in the ability to manage the combinatorial explosion inherent in infinite‑dimensional hypercomplex structures. By systematically collapsing redundant basis elements and then extracting independent sub‑algebras, one can obtain low‑dimensional models that retain essential multiplication characteristics of the original infinite system. Potential applications include high‑dimensional symmetry modeling in theoretical physics, where infinite‑dimensional Lie algebras often appear, and cryptographic constructions that benefit from complex multiplication rules but require efficient implementation.
Nevertheless, the authors acknowledge several limitations. First, the non‑canonical algebras may lack associativity or other stability properties, making numerical algorithms prone to error propagation. Second, the refactorization technique is only guaranteed for even dimensions; odd‑dimensional systems either cannot be halved or demand more elaborate transformations. Third, the choice of equivalence relation (i.e., the specific multiplication rule used for conversion) heavily influences the resulting algebra, and there is no general algorithm for selecting an “optimal” rule.
In conclusion, the paper contributes a novel algebraic framework for reducing infinite hypercomplex systems to manageable finite forms and for further halving the dimension of even‑dimensional outcomes. Future work should focus on formalizing the algebraic properties of the resulting non‑canonical structures, automating the refactorization process, and exploring concrete implementations in physics, engineering, and cryptography.