Lie Subalgebra

If g is a Lie algebra, then a subspace a of g is said to be a Lie subalgebra if it is closed under the Lie bracket. That is, a is a Lie subalgebra of g if for all x,y in a, it follows that [x,y] in a (where [·,·] is the Lie bracket in g).

For example, the vector space gl(n,C) of all n×n complex matrices is a Lie algebra with Lie bracket given by the matrix commutator: [X,Y]=XY-YX. The subspace sl(n,C) consisting of all traceless n×n complex matrices is a Lie subalgebra of gl(n,C) since the trace of a matrix commutator always vanishes.

A Lie subalgebra a is said to be an ideal of g if for all x in g and y in a, it follows that [x,y] in a. It is obvious that every Lie algebra g has at least two ideals: namely {0} and g itself. These ideals are considered "trivial." For a slightly better example: note that the subalgebra sl(n,C) is a non-trivial ideal of gl(n,C) if n>1.

See also

Lie Algebra

This entry contributed by Shawn Westmoreland

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Samelson, H. Notes on Lie Algebras. New York: Springer-Verlag, pp. 7-8, 1990.Wan, Z. Lie Algebras. Oxford: Pergamon Press, p. 4, 1975.

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Lie Subalgebra

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Westmoreland, Shawn. "Lie Subalgebra." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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