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  "body": "\n## Example\n\nThe group $SU(2)$ is the [[special unitary group|CZCWFPNS4N]] for $n = 2$.\nIts elements are exactly the matrices\n$$U = \\begin{pmatrix} a \u0026 -\\bar{b} \\\\ b \u0026 \\bar{a} \\end{pmatrix}, \\qquad a, b \\in \\C, \\quad \\abs{a}^2 + \\abs{b}^2 = 1.$$\nCheck: for such a matrix, $U^* U = \\begin{pmatrix} \\abs{a}^2 + \\abs{b}^2 \u0026 0 \\\\ 0 \u0026 \\abs{a}^2 + \\abs{b}^2 \\end{pmatrix} = I_2$ and $\\det U = \\abs{a}^2 + \\abs{b}^2 = 1$.\nConversely, a unitary matrix with determinant $1$ has its second column determined by its first column in this way.\n\nSo the map $U \\mapsto (a, b)$ identifies $SU(2)$ with the unit sphere $S^3 \\subset \\C^2 = \\R^4$.\nIt is a [[compact Lie group|9TFC6ADC1C]] of dimension $3$.\n\n## Remarks\n\nUnder the usual identification of $\\C^2$ with the [[quaternions|0GE4DVQKR8]], $SU(2)$ becomes isomorphic to the group of unit quaternions.\nSince $S^3$ is simply connected, $SU(2)$ is simply connected.\n$SU(2)$ acts on the trace-zero Hermitian $2 \\times 2$ matrices, a real vector space of dimension $3$, by $X \\mapsto U X U^*$.\nThis action preserves $-\\det X$, which is a positive definite quadratic form on that space.\nIt gives a surjective homomorphism $SU(2) \\to SO(3)$ onto the [[rotation group|263GGJFA21]] with kernel $\\set{\\pm I_2}$.\nSo $SU(2)$ is the universal cover of $SO(3)$, and it is also the spin group $\\mathrm{Spin}(3)$.\nIts Lie algebra $\\mathfrak{su}(2)$ has basis $i\\sigma_1, i\\sigma_2, i\\sigma_3$, where $\\sigma_1, \\sigma_2, \\sigma_3$ are the Pauli matrices.\n\n## Explanations\n\n### L1 human\n\n$SU(2)$ is one of the smallest non-commutative compact connected Lie groups.\nAs a space it is the $3$-sphere, the set of points at distance $1$ from the origin in four dimensions.\nEach element gives a rotation of ordinary space, and exactly two elements, $U$ and $-U$, give the same rotation.\nPhysicists use $SU(2)$ to describe the spin of particles such as the electron.\n",
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