Solvable Lie Group

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)

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 diagonal entries over the real numbers, e.g.,

(2)

such that for every . 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)
			(4)
			(5)

Any simply connected real solvable Lie group is diffeomorphic to Euclidean space. Without simple connectedness this can fail, as the circle Lie group shows. 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.