2019/08/12 by Joy Christian, Christian, Joy
Mathematics · #Advanced Topics in Algebra #Algebraic and Geometric Analysis #FOS: Mathematics #FOS: Physical sciences #General Mathematics (math.GM) #Homotopy and Cohomology in Algebraic Topology #Quantum Physics (quant-ph)
paper · pdf · doi:10.48550/arxiv.1908.06172
openalex publication_date 2019/08/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We present an eight-dimensional even sub-algebra of the 24=16-dimensional associative Clifford algebra Cl4,0 and show that its eight-dimensional multivectors \bf X and \bf Y respect the composition law ||\bf X\bf Y||=||\bf X|| ||\bf Y||, thus forming an octonion-like but associative normed division algebra, where the norms are calculated using the fundamental geometric product instead of the usual scalar product so that the underlying coefficient algebra resembles split complex numbers instead of reals. The corresponding 7-sphere obtained from projecting this multivector-valued composition law to the scalar-valued composition law has a topology that differs from that of the octonionic 7-sphere. Just as the octonionic 7-sphere is parallelizable using the non-associative algebra of octonions, we demonstrate that the 7-sphere presented herein is parallelizable using the said associative algebra.