Nilpotent Lie Group

A nilpotent Lie group is a Lie group G which is connected and whose Lie algebra is a nilpotent Lie algebra g. That is, its Lie algebra lower central series


eventually vanishes, g_k=0 for some k. 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.,

 [1 a_(12) a_(13); 0 1 a_(23); 0 0 1],

which is called the Heisenberg group. Its Lie algebra lower central series is given by

g_0=[0 b_(12) b_(13); 0 0 b_(23); 0 0 0]
g_1=[0 0 c_(13); 0 0 0; 0 0 0]
g_2=[0 0 0; 0 0 0; 0 0 0].

Any real nilpotent Lie group is diffeomorphic to Euclidean space. For instance, the group of matrices in the example above is diffeomorphic to R^3, 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.

See also

Borel Group, Group Representation, Lie Algebra, Lie Algebra Commutator Series, Lie Algebra Lower Central Series, Lie Group, Matrix, Nilpotent Lie Group Representation, Solvable Lie Algebra, Solvable Lie Group, Split Solvable Lie Algebra, Unipotent, Vector Space Flag

This entry contributed by Todd Rowland

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Rowland, Todd. "Nilpotent Lie Group." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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