A generalization of Euler's formula in Clifford algebras

摘要

In this paper, the equation ex∧y=1, such that x,y∈ℝn, is proved using defined group homomorphism and Euler's formula, where x ∧ y is the outer product between x and y. Firstly, this equation is verified for n=2,3,4,5 using Clifford product. Then, it turns out that it is difficult to maintain the proof in this way since the outer product is anticommutative, the size increases, and the calculation becomes a lot of work. For this reason, a group of homomorphism from the group Cℓn, 0 to the group ℝn,0 is described and used Euler's formula. Eventually, it is proved for any n∈ℤ+n≥2 by generalizing this equation.