The Lie Representation of the Complex Unit Sphere

TL;DR

This paper derives a 6-dimensional Lie group representation of the complex unit sphere, explores its structure and dynamics, and connects it to applications in physics and mathematics.

Contribution

It presents a novel derivation of the SO(3,C) Lie group and its unitary form U(3) for the six-dimensional complex sphere, advancing understanding of higher-dimensional division algebras.

Findings

Derived the 6D Eulerian Lie group SO(3,C)

Connected SO(3,C) to U(3) for the complex sphere

Potential applications in physics and complex analysis

Abstract

We present the derivation of the 6-dimensional Eulerian Lie group of the form SO(3,C). We describe our derivation process, which involves the creation of a finite group by using permutation matrices, and the exponentiation of the adjoint form of the subset representing the generators of the finite group. We take clues from the 2-dimensional complex rotation matrix to present, what we believe, is a true representation of the Lie group for the six-dimensional complex unit sphere and proceed to study its dynamics. With this approach, we can proceed to present this SO(3,C) group and derive its unitary counterpart that is U(3). The following findings can prove useful in mathematical physics, complex analysis and applications in deriving higher dimensional forms of similar division algebras.