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  "body": "\n## Example\n\nLet $n \\ge 1$ and let $\\mathbb{H}$ be the [[quaternions|0GE4DVQKR8]].\nFor a quaternionic matrix $A \\in M_n(\\mathbb{H})$, let $A^*$ be its quaternionic conjugate transpose.\nThe *compact symplectic group* is\n$$\\mathrm{Sp}(n) = \\set{A \\in M_n(\\mathbb{H}) : A^* A = I_n},$$\nthe group of $\\mathbb{H}$-linear maps of $\\mathbb{H}^n$ that preserve the quaternionic inner product $\\ip{x}{y} = \\sum_k \\bar{x}_k y_k$.\nIt is also called the quaternionic unitary group.\n\nFor $n = 1$, the condition reads $\\bar{q} q = 1$, so $\\mathrm{Sp}(1)$ is the group of unit quaternions, which is isomorphic to [[SU(2)|E5P8AG6CK7]].\n\nWriting $\\mathbb{H}^n = \\C^{2n}$ turns $\\mathrm{Sp}(n)$ into a group of complex $2n \\times 2n$ matrices, and then\n$$\\mathrm{Sp}(n) = \\mathrm{Sp}(2n, \\C) \\cap U(2n),$$\nthe intersection of the complex [[symplectic group|X12FS2JAP0]] with the [[unitary group|0R9R0Y1HDK]].\nThis is a closed and bounded subgroup of $GL_{2n}(\\C)$, so $\\mathrm{Sp}(n)$ is a [[compact Lie group|9TFC6ADC1C]].\n\n## Remarks\n\n$\\mathrm{Sp}(n)$ is connected and simply connected, and its real dimension is $n(2n + 1)$.\nIts Lie algebra is simple of type $C_n$.\nIt is a maximal compact subgroup of $\\mathrm{Sp}(2n, \\C)$.\nThe maximal compact subgroup of the real group $\\mathrm{Sp}(2n, \\R)$ is different: it is isomorphic to $U(n)$.\nThe notation $\\mathrm{Sp}(n)$ for this group clashes with the notation $\\mathrm{Sp}(2n)$ for the noncompact symplectic groups, and some authors write $\\mathrm{Sp}(2n)$ or $USp(2n)$ for it.\n\n## Explanations\n\n### L1 human\n\nThe compact symplectic group plays for the quaternions the role that the orthogonal group plays for the real numbers and the unitary group plays for the complex numbers.\nIt collects the quaternionic matrices that keep the length of every vector.\nFor $n = 1$ it is the set of quaternions of length $1$, which is a $3$-sphere.\nTogether with the special orthogonal and special unitary groups it gives the infinite families of compact simple Lie groups.\n",
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