Solvable Lie Group

A solvable Lie group is a Lie group G which is connected and whose Lie algebra g is a solvable Lie algebra. That is, the Lie algebra commutator series

g^1=[g,g],g^2=[g^1,g^1],...

(1)

eventually vanishes, g^k=0 for some k. 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.,

[画像: [a_(11) a_(12) a_(13); 0 a_(22) a_(23); 0 0 a_(33)] ]

(2)

such that product_(i)a_(ii)>0. The Lie algebra g of G 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

g^1 = [画像:[0 b_(12) b_(13); 0 0 b_(23); 0 0 0]]

(3)

g^2 = [画像:[0 0 c_(13); 0 0 0; 0 0 0]]

(4)

g^3 = [画像:[0 0 0; 0 0 0; 0 0 0].]

(5)

Any real solvable Lie group is diffeomorphic to Euclidean space. For instance, the group of matrices in the example above is diffeomorphic to R^6, via the Lie group exponential map. However, in general, the exponential map in a solvable Lie algebra need not be surjective.

See also

Group Representation, Lie Algebra, Lie Algebra Commutator Series, Lie Group, Matrix, Nilpotent Lie Group, Solvable Group, Solvable Lie Algebra, Vector Space Flag

This entry contributed by Todd Rowland

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Rowland, Todd. "Solvable Lie Group." From MathWorld--A Wolfram Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/SolvableLieGroup.html

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