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  "body": "\n## Example\n\nThe table lists common real [[Lie groups|7JBRGCVJRB]] with a few invariants.\nThese are the real dimension, [[compactness|0TVJZYB6SH]], the number of connected components, and the [[fundamental group|S1XK3HXHFX]] $\\pi_1$ of the identity component.\nHere $n \\ge 3$, to keep the entries uniform.\n\n| Group | Dimension | Compact | Components | $\\pi_1$ |\n|---|---|---|---|---|\n| $\\R^n$ | $n$ | no | $1$ | $0$ |\n| $U(1)$ | $1$ | yes | $1$ | $\\Z$ |\n| $GL_n(\\R)$ | $n^2$ | no | $2$ | $\\Z/2$ |\n| $SL_n(\\R)$ | $n^2 - 1$ | no | $1$ | $\\Z/2$ |\n| $O(n)$ | $n(n-1)/2$ | yes | $2$ | $\\Z/2$ |\n| $SO(n)$ | $n(n-1)/2$ | yes | $1$ | $\\Z/2$ |\n| $\\mathrm{Spin}(n)$ | $n(n-1)/2$ | yes | $1$ | $0$ |\n| $U(n)$ | $n^2$ | yes | $1$ | $\\Z$ |\n| $SU(n)$ | $n^2 - 1$ | yes | $1$ | $0$ |\n| $GL_n(\\C)$ | $2n^2$ | no | $1$ | $\\Z$ |\n| $SL_n(\\C)$ | $2n^2 - 2$ | no | $1$ | $0$ |\n| $Sp(2n, \\R)$ | $n(2n+1)$ | no | $1$ | $\\Z$ |\n| $Sp(n)$ | $n(2n+1)$ | yes | $1$ | $0$ |\n\nThe rows are the [[circle group|04EAFVFTF7]] $U(1)$, the [[general linear group|0FCHE902D7]] and the [[special linear group|0V9PMN0RBH]].\nThen come the [[orthogonal group|0TKYAEJ8N7]], the [[special orthogonal group|0CGYNDCWRR]], the [[spin group|JJSXXJQGEH]], the [[unitary group|0R9R0Y1HDK]] and the [[special unitary group|CZCWFPNS4N]].\nThe last rows are the real [[symplectic group|X12FS2JAP0]] $Sp(2n, \\R)$ of $2n \\times 2n$ real matrices.\nThey also include the [[compact symplectic group|5NNPBFQA02]] $Sp(n)$ of quaternionic $n \\times n$ matrices.\n\nCheck of the row $O(n)$.\nIts Lie algebra is the space of real skew-symmetric $n \\times n$ matrices, which has dimension $n(n-1)/2$.\nThe determinant of an orthogonal matrix is $\\pm 1$, and both values occur, so $O(n)$ has at least two components.\nEach fibre of $\\det$ is a translate of the connected group $SO(n)$, so there are exactly two.\n$O(n)$ is closed and bounded in $\\R^{n^2}$, since $A^T A = I$ forces every entry into $[-1, 1]$.\nSo $O(n)$ is compact.\n\n## Remarks\n\nFor $n = 2$ some entries change: $SO(2) \\cong U(1)$ and the identity components of $GL_2(\\R)$ and $SL_2(\\R)$ have fundamental group $\\Z$.\nThe $\\pi_1$ entries for the noncompact groups use one fact.\nA connected real Lie group deformation retracts onto a maximal compact subgroup, for example $SO(n)$ in $SL_n(\\R)$ and $U(n)$ in $Sp(2n, \\R)$.\nFor $n \\ge 3$, $\\mathrm{Spin}(n)$ is the universal cover of $SO(n)$, a double cover.\n\n## Explanations\n\n### L1 human\n\nMost Lie groups met in practice are groups of matrices: invertible matrices, matrices of determinant one, rotations, unitary matrices and symplectic matrices.\nThe table records a few basic numbers for each one.\nThe dimension counts how many independent parameters describe an element, as the three angles of a rotation in space.\nCompactness says whether the group is bounded, and the components and the fundamental group describe its shape as a space.\n\n### L2 human\n\nThe invariants in the table are linked through maximal compact subgroups.\nA connected Lie group has the homotopy type of its maximal compact subgroup, so $\\pi_1$ of $SL_n(\\R)$ is read off from $SO(n)$ and $\\pi_1$ of $Sp(2n, \\R)$ from $U(n)$.\nMany rows pair a noncompact group with a compact form of the same complexified Lie algebra, such as $SL_n(\\R)$ and $SU(n)$.\nThis explains why such pairs, like $Sp(2n, \\R)$ and $Sp(n)$, have the same dimension.\n",
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