SU2Algebra

SU2 algebra, formally denoted su(2), is the Lie algebra of the special unitary group SU(2). It is a real, three-dimensional, simple and compact Lie algebra. A standard basis {T1, T2, T3} satisfies the commutation relations [T_a, T_b] = i ε_{abc} T_c, where ε_{abc} is the Levi-Civita symbol. Equivalently, in an anti-Hermitian basis X_a = i T_a one has [X_a, X_b] = ε_{abc} X_c. In the fundamental 2x2 representation, one may take T_a = σ_a/2, with σ_a the Pauli matrices. The generators can be identified with angular momentum operators J_i, obeying [J_i, J_j] = i ε_{ijk} J_k.

su(2) is isomorphic to so(3) as real Lie algebras; the groups SU(2) and SO(3) share the same Lie algebra, but SU(2) is the universal covering group of SO(3). Finite-dimensional irreducible representations of su(2) are labeled by a nonnegative half-integer j (spin), with dimension 2j+1. Integer j correspond to representations that descend to SO(3), while half-integer j yield projective representations of SO(3) but true representations of SU(2).

The Killing form on su(2) is negative definite; with the T_a = σ_a/2 normalization, it is proportional to δ_{ab}, often K(T_a, T_b) = −2 δ_{ab}. The algebra is simple, of rank 1, and its root system is of type A1.

Applications of su(2) are widespread: it underpins the description of spin and angular momentum in quantum mechanics and forms the gauge algebra of SU(2) in the electroweak sector of the Standard Model, alongside U(1).