Complete purely algebraic proof of the homomorphism between SU(2) and SO(3) without concerning their topological properties

Muhammad Ardhi Khalif (Universitas Islam Negeri Walisongo Semarang)
Nur Farida Amalia (Universitas Diponegoro)

Journal of Natural Sciences and Mathematics Research
Vol 8, No 2 (2022): December

The aim of this paper is to provide a complete purely algebraic proof of homo-morphism between SU (2) and SO(3) without concerning the topology of bothgroups. The proof is started by introducing a map ϕ : SU (2) → M L(3, C) de-fined as [ϕ(U )] i j ≡ 12 tr(σ i U σ j U † ). Firstly we proof that the map ϕ satisfies[ϕ(U 1 U 2 )] i j = [ϕ(U 1 )] i k [ϕ(U 2 )] k j , for every U 1 , U 2 ∈ SU (2). The next step is toshow that the collection of ϕ(U ) is having orthogonal property and every ϕ(U ) hasdeterminant of 1. After that, we proof that ϕ(I 2 ) = I 3 . Finally, to make sure thatϕ is indeed a homomorphism, not an isomorphism, we proof that ϕ(−U ) = ϕ(U ),∀U ∈ SU (2).