The Lie algebra su(2)

If we have a vector space of operators spanned by \(\mathfrak{g} = \set{T^x, T^y, T^z}\) with the commutation relations \[\begin{align} \require{physics} & \comm{T^x}{T^y} = i T^z \\ & \comm{T^y}{T^z} = i T^x \\ & \comm{T^z}{T^x} = i T^y \thinspace , \end{align}\] the algebra \(\mathfrak{g}\) is said to be \(\mathfrak{su}(2)\). Another basis for \(\mathfrak{su}(2)\) could be given by \[\begin{align} & T^+ = T^x + i T^y \\ & T^- = T^x - i T^y \\ & T^z = T^z \thinspace , \end{align}\] or, reversely \[\begin{align} & T^x = \frac{1}{2} \qty(T^+ + T^-) \\ & T^y = \frac{1}{2i} \qty(T^+ - T^-) \\ & T^z = T^z \thinspace , \end{align}\] with the commutators \[\begin{align} & \comm{T^+}{T^-} = 2 T^z \\ & \comm{T^z}{T^\pm} = \pm T^\pm \thinspace , \end{align}\] and \(T^+\) being the Hermitian adjoint of \(T^-\) \[\begin{equation} T^- = \qty(T^+)^\dagger \thinspace , \end{equation}\] and \(T^z\) being Hermitian: \[\begin{equation} \qty(T^z)^\dagger = T^z \thinspace . \end{equation}\] The (quadratic) Casimir invariant of \(\mathfrak{su}(2)\) is given by \[\begin{align} T^2 &= \qty(T^x)^2 + \qty(T^y)^2 + \qty(T^z)^2 \\ &= T^+ T^- - T^z + \qty(T^z)^2 \thinspace , \end{align}\] and it commutes with every generator: \[\begin{align} & \comm{T^2}{T^x} = \comm{T^2}{T^y} = \comm{T^2}{T^z} = 0 \\ & \comm{T^2}{T^+} = \comm{T^2}{T^-} = \comm{T^2}{T^z} = 0 \thinspace . \end{align}\]

If we were to take \(N\) copies of the \(\mathfrak{su}(2)\)-algebra, the following commutation relations hold: \[\begin{align} & \comm{T^+_i}{T^-_j} = 2 \delta_{ij} T^z_i \\ & \comm{T^z_i}{T^\pm_j} = \pm \delta_{ij} T^\pm_i \thinspace . \end{align}\]