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Furthermore, every rotation arises from exactly two versors in this fashion. In short: there is a 2:1 surjective homomorphism from SU(2) to SO(3); consequently SO(3) is isomorphic to the quotient group SU(2)/{±I}, the manifold underlying SO(3) is obtained by identifying antipodal points of the 3-sphere S 3, and SU(2) is the universal cover of ...
The representation with = (i.e., = / in the physics convention) is the 2 representation, the fundamental representation of SU(2). When an element of SU(2) is written as a complex 2 × 2 matrix, it is simply a multiplication of column 2-vectors.
These matrices are traceless, Hermitian, and obey the extra trace orthonormality relation, so they can generate unitary matrix group elements of SU(3) through exponentiation. [1] These properties were chosen by Gell-Mann because they then naturally generalize the Pauli matrices for SU(2) to SU(3), which formed the basis for Gell-Mann's quark ...
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 } .
Thus one can define a (unique) unitary group of dimension n for the extension F q 2 /F q, denoted either as U(n, q) or U(n, q 2) depending on the author. 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).
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.
= ¯ (′) where B μ is the U(1) gauge field; Y W is the weak hypercharge (the generator of the U(1) group); W μ is the three-component SU(2) gauge field; and the components of τ are the Pauli matrices (infinitesimal generators of the SU(2) group) whose eigenvalues give the weak isospin.
Thus, for example, the adjoint representation of su(2) is the defining representation of so(3). Examples.