A nilpotent Lie group is a Lie group which is connected and whose Lie algebra is a nilpotent Lie algebra . That is, its Lie algebra lower central series
(1)
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eventually vanishes, for some . So a nilpotent Lie group is a special case of a solvable Lie group.
The basic example is the group of upper triangular matrices with 1s on their diagonals, e.g.,
(2)
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which is called the Heisenberg group. Its Lie algebra lower central series is given by
(3)
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(4)
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(5)
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Any real nilpotent Lie group is diffeomorphic to Euclidean space. For instance, the group of matrices in the example above is diffeomorphic to , via the Lie group exponential map. In general, the exponential map of a nilpotent Lie algebra is surjective, in contrast to the more general solvable Lie group.