A solvable Lie group is a Lie group which is connected and whose Lie algebra is a solvable Lie algebra. That is, the Lie algebra commutator series
(1)
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eventually vanishes, for some . Since nilpotent Lie algebras are also solvable, any nilpotent Lie group is a solvable Lie group.
The basic example is the group of invertible upper triangular matrices with positive determinant, e.g.,
(2)
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such that . The Lie algebra of is its tangent space at the identity matrix, which is the vector space of all upper triangular matrices, and it is a solvable Lie algebra. Its Lie algebra commutator series is given by
(3)
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(4)
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(5)
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Any real solvable 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. However, in general, the exponential map in a solvable Lie algebra need not be surjective.