Nilpotent Lie algebra


Nilpotent Lie algebra

In mathematics, a Lie algebra \mathfrak{g} is nilpotent if the lower central series

 \mathfrak{g} > [\mathfrak{g},\mathfrak{g}] > [[\mathfrak{g},\mathfrak{g}],\mathfrak{g}] > [[[\mathfrak{g},\mathfrak{g}],\mathfrak{g}],\mathfrak{g}] > \cdots

becomes zero eventually. Equivalently, \mathfrak{g} is nilpotent if

\operatorname{ad}(x_1) \operatorname{ad}(x_2) \operatorname{ad}(x_3) ... \operatorname{ad}(x_r) = 0

for any sequence xi of elements of \mathfrak{g} of sufficiently large length. (Here, \operatorname{ad}(x) is given by \operatorname{ad}(x)y = [x, y].) Consequences are that \operatorname{ad}(x) is nilpotent (as a linear map), and that the Killing form of a nilpotent Lie algebra is identically zero. (In comparison, a Lie algebra is semisimple if and only if its Killing form is nondegenerate.)

Every nilpotent Lie algebra is solvable; this fact gives one of the powerful ways to prove the solvability of a Lie algebra since, in practice, it is usually easier to prove the nilpotency than the solvability. The converse is not true in general. A Lie algebra \mathfrak{g} is nilpotent if and only if its quotient over an ideal containing the center of \mathfrak{g} is nilpotent.

Most of classic classification results on nilpotency are concerned with finite-dimensional Lie algebras over a field of characteristic 0. Let \mathfrak{g} be a finite-dimensional Lie algebra. \mathfrak{g} is nilpotent if and only if \operatorname{ad}(\mathfrak{g}) is nilpotent. Engel's theorem states that \mathfrak{g} is nilpotent if and only if \operatorname{ad}(x) is nilpotent for every x \in \mathfrak{g}. \mathfrak{g} is solvable if and only if [\mathfrak{g}, \mathfrak{g}] is nilpotent.

Examples

  • Every subalgebra and quotient of a nilpotent Lie algebra is nilpotent.
  • If \mathfrak{gl}_k is the set of k\times k matrices, then the subalgebra consisting of strictly upper triangular matrices, denoted by \mathfrak{n}_k, is a nilpotent Lie algebra.
  • A Heisenberg algebra is nilpotent.
  • A Cartan subalgebra of a Lie algebra is nilpotent and self-normalizing.

References

  • Humphreys, James E. Introduction to Lie Algebras and Representation Theory. Graduate Texts in Mathematics, 9. Springer-Verlag, New York, 1972. ISBN 0-387-90053-5

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