The Classification and Geometric Interpretations of Hyperbolic Spinors Related to Split Quaternions

Emrah Yıldırım; Tülay Erişir

The Classification and Geometric Interpretations of Hyperbolic Spinors Related to Split Quaternions

Abstract

This study investigates the relationship between split quaternions and hyperbolic spinors by considering the linear correspondence that assigns a hyperbolic spinor to each split quaternion. Based on this correspondence, hyperbolic spinors are classified by taking into account the algebraic classification of the associated split quaternions. In this way, different types of hyperbolic spinors are identified according to the character of the corresponding split quaternion. Using this relation, the hyperbolic spinor representations of the left and right multiplication matrices of split quaternions are explicitly constructed. We obtain these matrices and derive several identities describing their algebraic properties. Moreover, we show that the left hyperbolic spinor matrix plays a fundamental role within this structure. Its eigenvalues are computed and these values are classified according to the type of the corresponding hyperbolic spinors. The results provide a clear algebraic and geometric description of hyperbolic spinors arising from split quaternionic structures. This approach is expected to contribute to geometric algebra, differential geometry, algebra and relativistic physics where spinor representations are essential.

Keywords

Hyperbolic spinors, split quaternions, hyperbolic spinor matrices

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Details

Primary Language

English

Subjects

Algebraic and Differential Geometry

Journal Section

Research Article

Authors

Emrah Yıldırım
0000-0001-8993-0447
Türkiye

Tülay Erişir ^*
0000-0001-6444-1460
Türkiye

Early Pub Date

June 5, 2026

Publication Date

-

Submission Date

April 2, 2026

Acceptance Date

May 18, 2026

Published in Issue

Year 2026 Number: Advanced Online Publication

IZ

https://izlik.org/JA35EA25ZX

APA

Yıldırım, E., & Erişir, T. (2026). The Classification and Geometric Interpretations of Hyperbolic Spinors Related to Split Quaternions. Sakarya Journal of Mathematics, Advanced Online Publication, 22-36. https://izlik.org/JA35EA25ZX

AMA

1.Yıldırım E, Erişir T. The Classification and Geometric Interpretations of Hyperbolic Spinors Related to Split Quaternions. Sakarya Journal of Mathematics. 2026;(Advanced Online Publication):22-36. https://izlik.org/JA35EA25ZX

Chicago

Yıldırım, Emrah, and Tülay Erişir. 2026. “The Classification and Geometric Interpretations of Hyperbolic Spinors Related to Split Quaternions”. Sakarya Journal of Mathematics, no. Advanced Online Publication: 22-36. https://izlik.org/JA35EA25ZX.

EndNote

Yıldırım E, Erişir T (June 1, 2026) The Classification and Geometric Interpretations of Hyperbolic Spinors Related to Split Quaternions. Sakarya Journal of Mathematics Advanced Online Publication 22–36.

IEEE

[1]E. Yıldırım and T. Erişir, “The Classification and Geometric Interpretations of Hyperbolic Spinors Related to Split Quaternions”, Sakarya Journal of Mathematics, no. Advanced Online Publication, pp. 22–36, June 2026, [Online]. Available: https://izlik.org/JA35EA25ZX

ISNAD

Yıldırım, Emrah - Erişir, Tülay. “The Classification and Geometric Interpretations of Hyperbolic Spinors Related to Split Quaternions”. Sakarya Journal of Mathematics. Advanced Online Publication (June 1, 2026): 22-36. https://izlik.org/JA35EA25ZX.

JAMA

1.Yıldırım E, Erişir T. The Classification and Geometric Interpretations of Hyperbolic Spinors Related to Split Quaternions. Sakarya Journal of Mathematics. 2026;:22–36.

MLA

Yıldırım, Emrah, and Tülay Erişir. “The Classification and Geometric Interpretations of Hyperbolic Spinors Related to Split Quaternions”. Sakarya Journal of Mathematics, no. Advanced Online Publication, June 2026, pp. 22-36, https://izlik.org/JA35EA25ZX.

Vancouver

1.Emrah Yıldırım, Tülay Erişir. The Classification and Geometric Interpretations of Hyperbolic Spinors Related to Split Quaternions. Sakarya Journal of Mathematics [Internet]. 2026 Jun. 1;(Advanced Online Publication):22-36. Available from: https://izlik.org/JA35EA25ZX