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  "body": "\n## Summary\n\nThe universal covering group of a connected [[Lie group|7JBRGCVJRB]] $G$ is its [[universal cover|Y1YPE6PM52]] $\\tilde G$ with a Lie group structure.\nThis structure makes the covering map $\\tilde G \\to G$ a homomorphism.\nIt is simply connected, and $G$ is the quotient of $\\tilde G$ by a discrete central subgroup isomorphic to the [[fundamental group|S1XK3HXHFX]] of $G$.\nAll connected Lie groups with a given Lie algebra are quotients of the same universal covering group.\n\n## Explanations\n\n### L1 human\n\nSome Lie groups have loops that cannot be shrunk, like the circle.\nThe universal covering group unwinds all such loops while keeping a group structure.\nFor the circle of unit complex numbers, it is the real line, mapped onto the circle by $t \\mapsto e^{2\\pi i t}$; adding real numbers matches multiplying points of the circle.\nFor the rotations of space $SO(3)$, it is the group $SU(2)$, and each rotation is covered by exactly two elements.\n",
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  "source": "https://github.com/tamnd/math2/blob/main/nodes/SWW/SWW2993CVJ.md",
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