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  "body": "\n## Definition\n\nLet $G$ be a [[connected|0V032VWTHD]] [[Lie group|7JBRGCVJRB]].\nA *universal covering group* of $G$ is a Lie group $\\tilde G$ together with a map $p \\colon \\tilde G \\to G$ such that:\n\n1. $\\tilde G$ is connected and simply connected, that is, its [[fundamental group|S1XK3HXHFX]] is trivial;\n2. $p$ is a homomorphism of Lie groups;\n3. $p$ is a [[covering map|3XZRRE165N]].\n\n## Remarks\n\nExistence: let $p \\colon \\tilde G \\to G$ be the [[universal cover|Y1YPE6PM52]] of the manifold $G$, and choose $\\tilde e$ over the identity $e$.\nThere is a unique Lie group structure on $\\tilde G$ with identity $\\tilde e$ for which $p$ is a homomorphism.\nThe universal covering group is unique up to isomorphism over $G$.\nThe kernel of $p$ is a discrete central subgroup of $\\tilde G$, isomorphic to $\\pi_1(G)$.\n\nExample: $p \\colon \\R \\to S^1$, $p(t) = e^{2\\pi i t}$, is a homomorphism from $(\\R, +)$, a covering map, and $\\R$ is simply connected; the kernel is $\\Z = \\pi_1(S^1)$.\n\nExample: identify $SU(2)$ with the unit quaternions and $\\R^3$ with the purely imaginary quaternions.\nThe map sending $q$ to the rotation $v \\mapsto q v q^{-1}$ is a universal covering group $SU(2) \\to SO(3)$ with kernel $\\set{\\pm 1}$.\n\nNon-example: the identity map $SO(3) \\to SO(3)$ is a homomorphism and a covering map, but $SO(3)$ is not simply connected, since $\\pi_1(SO(3)) \\cong \\Z/2$.\n",
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