$\mathrm{SU}(2)$ is an [[SU(n)]] group where $n=2.$ Thus it contains the set of $2\times2$ [unitary matrices](Unitary%20operators.md#Unitary%20matrices) with a [determinant](Determinants.md#Determinant%20of%20a%20matrix) of $1$. %% It would be cool to elaborate the formal relationship that exists between SU(n) groups and SU(2), SU(3), etc.%% # $\mathrm{SU}(2)$ Representation The [[Lie group representations]]s take the following form: $\mbox{SU}(2) = \begin{Bmatrix}\begin{pmatrix}\alpha & -\beta^*\\ \beta & \alpha^* \end{pmatrix}:\; \alpha, \beta \in \mathbb{C},\;\; |\alpha|^2 + |\beta|^2 = 1\end{Bmatrix}$ where $\alpha$ and $\beta$ are arbitrary complex numbers. This is proven with a direct calculation [here.](SU(2).md#Proof%20of%20its%20representation) # Group Algebra of $\mathrm{SU}(2)$ [Lie algebras](Lie%20algebras.md) # Group Topology of $\mathrm{SU}(2)$ $\mathrm{SU}(2)$ is a [Simply connected Lie groups](Simply%20connected%20Lie%20groups.md). --- # Proofs and Examples ## Proof of its representation It is sufficient to show that all $2\times2$ matrices with the above form where $|\alpha|^2 + |\beta|^2 = 1$ are unitary. Thus we multiply the matrix by its [conjugate transpose](Adjoint.md#conjugate%20transpose) to see that we obtain the identity matrix: $\begin{pmatrix}\alpha & -\beta^*\\ \beta & \alpha^* \end{pmatrix}\begin{pmatrix}\alpha & -\beta^*\\ \beta & \alpha^* \end{pmatrix}^\dagger = \begin{pmatrix}\alpha & -\beta^*\\ \beta & \alpha^* \end{pmatrix}\begin{pmatrix}\alpha^* & \beta^*\\ -\beta & \alpha \end{pmatrix}=\begin{pmatrix}\alpha\alpha^*+\beta\beta^* & \beta^*\alpha - \beta^*\alpha\\ \alpha^*\beta-\alpha^*\beta & \beta^*\beta+\alpha^*\alpha \end{pmatrix}$ $=\begin{pmatrix}1 & 0\\0 & 1 \end{pmatrix}$ #MathematicalFoundations/Algebra/AbstractAlgebra/GroupTheory/Lie/LieGroups/Algebras/LieAlgebras #MathematicalFoundations/Algebra/AbstractAlgebra/GroupTheory/Lie/LieGroups