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  "body": "\n## Summary\n\nA nilpotent Lie group is a connected [[Lie group|7JBRGCVJRB]] that is a [[nilpotent group|0Q7MMQHC1P]].\nSo its lower central series reaches the trivial group after finitely many steps.\nEquivalently, its Lie algebra is a nilpotent Lie algebra.\nEvery nilpotent Lie group is a [[solvable Lie group|9CNSVFQ801]], but not conversely.\nFor a connected and simply connected nilpotent Lie group, the exponential map from the Lie algebra is a diffeomorphism.\nSo such a group can be studied almost entirely through its Lie algebra.\n\n## Explanations\n\n### L1 human\n\nA nilpotent Lie group is close to being commutative.\nThe commutators of its elements may be nontrivial, but commutators with commutators, repeated a fixed number of times, are always trivial.\nThe basic example is the Heisenberg group of upper triangular $3 \\times 3$ matrices with $1$ on the diagonal.\nThere, any two elements commute up to a central element, and central elements commute with everything.\nGroups of this kind model the position and momentum operators of quantum mechanics.\n",
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