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The group SU(2) is isomorphic to the group of quaternions of norm 1, and is thus diffeomorphic to the 3-sphere. Since unit quaternions can be used to represent rotations in 3-dimensional space (up to sign), there is a surjective homomorphism from SU(2) to the rotation group SO(3) whose kernel is {+I, −I}.
SU(2) is the universal covering group of SO(3), and so its representation theory includes that of the latter, by dint of a surjective homomorphism to it. This underlies the significance of SU(2) for the description of non-relativistic spin in theoretical physics ; see below for other physical and historical context.
The analogue for a finite subgroup of Γ of SU(2) is the finite-dimensional group algebra C[Γ] From the Clebsch-Gordan rules, the convolution algebra A is isomorphic to a direct sum of n × n matrices, with n = 2j + 1 and j ≥ 0. The matrix coefficients for each irreducible representation V j form a set of matrix units.
The group SU(2) is the Lie group of unitary 2 × 2 matrices with unit determinant; its Lie algebra is the set of all 2 × 2 anti-Hermitian matrices with trace 0. Direct calculation, as above, shows that the Lie algebra s u 2 {\displaystyle {\mathfrak {su}}_{2}} is the three-dimensional real algebra spanned by the set { iσ k } .
The subgroup of the unitary group consisting of matrices of determinant 1 is called the special unitary group and denoted SU(n, q) or SU(n, q 2). For convenience, this article will use the U( n , q 2 ) convention.
n(2n+1) U(n) unitary group: complex n×n unitary matrices: Y 0 Z: R×SU(n) For n=1: isomorphic to S 1. Note: this is not a complex Lie group/algebra u(n) n 2: SU(n) special unitary group: complex n×n unitary matrices with determinant 1 Y 0 0 Note: this is not a complex Lie group/algebra su(n) n 2 −1
The group Spin(3) is isomorphic to the special unitary group SU(2); it is also diffeomorphic to the unit 3-sphere S 3 and can be understood as the group of versors (quaternions with absolute value 1). The connection between quaternions and rotations, commonly exploited in computer graphics, is explained in quaternions and spatial rotations.
The electroweak sector is a Yang–Mills gauge theory with the symmetry group U(1) × SU(2) L, = ¯ + ¯ + ¯ + ¯ + ¯, where the subscript sums over the three generations of fermions; ,, and are the left-handed doublet, right-handed singlet up type, and right handed singlet down type quark fields; and and are the left-handed doublet and right ...