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

Muhammad Ardhi Khalif; Nur Farida Amalia

doi:10.21580/jnsmr.2022.8.2.17519

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

* - Universitas Islam Negeri Walisongo Semarang, Indonesia

- Universitas Diponegoro, Indonesia

(*) Corresponding Author

DOI: 10.21580/jnsmr.2022.8.2.17519

Abstract
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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 both
groups. 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 to
show that the collection of ϕ(U ) is having orthogonal property and every ϕ(U ) has
determinant of 1. After that, we proof that ϕ(I 2 ) = I 3 . Finally, to make sure that
ϕ is indeed a homomorphism, not an isomorphism, we proof that ϕ(−U ) = ϕ(U ),
∀U ∈ SU (2).

Keywords: homomorphism; rotation group; orthogonal group; topological properties;

M Andrews and J Gunson. “Complex angular momenta and many-particle states. I. properties of local representations of the rotation group”. en. In: J. Math. Phys. 5.10 (1964), pp. 1391–1400.
V Bargmann. “On the representations of the rotation group”. en. In: Rev. Mod. Phys. 34.4 (1962), pp. 829–845.
Johan G Belinfante and Bernard Kolman. A survey of lie groups and lie algebras with applications and computational methods. Society for Industrial and Applied Mathematics, 1989.
Moshe Carmeli. “Representations of the threedimensional rotation group in terms of direction and angle of rotation”. en. In: J. Math. Phys. 9.11 (1968), pp. 1987–1992.
John F Cornwell. Group Theory in Physics: Volume 1: An Introduction. en. San Diego, CA: Academic Press, 1997.
Veliko Donchev, Clementina Mladenova, and Ivailo Mladenov. “On Vector-Parameter Form of the SU(2) → SO(3,R) Map”. In: Ann. Sofia Univ., Fac. Math and Inf. 102 (Sept. 2015), pp. 91–107.
Robert Gilmore. Lie groups, lie algebras & some of their applications. Mineola, NY: Dover Publications, 2012.
Erik W Grafarend and Wolfgang K¨uhnel. “A minimal atlas for the rotation group SO(3)”. en. In: GEM Int. J. Geomath. 2.1 (2011), pp. 113-122.
Brian C Hall. Lie groups, lie algebras, and representations: An elementary introduction. en. 2nd ed. Cham, Switzerland: Springer International Publishing, 2016.
Francis Halzen and Alan D Martin. Quarks and leptons: Introductory course in modern particle physics. Brisbane, QLD, Australia: John Wiley and Sons (WIE), 1984.
J Dwayne Hamilton. “The rotation and precession of relativistic reference frames”. en. In: Can. J. Phys. 59.2 (1981), pp. 213–224.
Francesco Iachello. Lie Algebras and Applications. en. Berlin, Germany: Springer, 2015.
James D Louck. “From the rotation group to the Poincar´e group”. In: Symmetries in Science VI. Boston, MA: Springer US, 1993, pp. 455–468.
Franklin Lowenthal. “Uniform finite generation of the rotation group”. In: Rocky Mountain J. Math. 1.4 (1971), pp. 575–586.
Chi-Sing Man. “Irreducible Representations of SU(2), SO(3), and O(3)”. In: Crystallographic Texture and Group Representations. Dordrecht: Springer Netherlands, 2023, pp. 313–328.
H Maschke. “The representation of finite groups, especially of the rotation groups of the regular bodies of three-and four-dimensional space, by cayley’s color diagrams”. In: Amer. J. Math. 18.2 (1896), p. 156.
Clair E Miller. “The topology of rotation groups”. In: Ann. Math. 57.1 (1953), p. 90.
C F Moppert. “The group of rotations of the sphere: Generators and relations”. en. In: Geom. Dedicata 18.1 (1985).
F.D. Murnaghan. Lectures on Applied Mathematics: The unitary and rotation groups. Lectures on Applied Mathematics. Spartan Books, 1962. url: https://books.google.co.id/ books?id=05ezAAAAIAAJ.
Alyssa Novelia and Oliver M O’Reilly. “On geodesics of the rotation group SO(3)”. en. In: Regul. Chaotic Dyn. 20.6 (2015), pp. 729–738.
R Raimondi et al. “Spin-orbit interaction in a two-dimensional electron gas: A SU(2) formulation”. en. In: Ann. Phys. 524.3-4 (2012).
Miles Reid and Balazs Szendroi. “Quaternions, rotations and the geometry of transformation groups”. In: Geometry and Topology. Cambridge: Cambridge University Press, 2005, pp. 142–163.
D H Sattinger and O L Weaver. Lie groups and algebras with applications to physics, geometry, and mechanics. en. 1986th ed. New York, NY: Springer, 2013.
Wu-Ki Tung. Group Theory in Physics. World Scientific Publishing, 1985.
Brian G Wybourne. Classical Groups for Physicists. en. Nashville, TN: John Wiley & Sons, 1974.