{"title":"Complete purely algebraic proof of the homomorphism between SU(2) and SO(3) without concerning their topological properties","authors":"Muhammad Ardhi Khalif, Nur Farida Amalia","doi":"10.21580/jnsmr.2022.8.2.17519","DOIUrl":"https://doi.org/10.21580/jnsmr.2022.8.2.17519","url":null,"abstract":"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).","PeriodicalId":191192,"journal":{"name":"Journal of Natural Sciences and Mathematics Research","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114508164","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}