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Concept
Compact Lie algebra
Summary
In the mathematical field of Lie theory, there are two definitions of a compact Lie algebra. Extrinsically and topologically, a compact Lie algebra is the Lie algebra of a compact Lie group; this definition includes tori. Intrinsically and algebraically, a compact Lie algebra is a real Lie algebra whose Killing form is negative definite; this definition is more restrictive and excludes tori,. A compact Lie algebra can be seen as the smallest real form of a corresponding complex Lie algebra, namely the complexification.
Formally, one may define a compact Lie algebra either as the Lie algebra of a compact Lie group, or as a real Lie algebra whose Killing form is negative definite. These definitions do not quite agree:
The Killing form on the Lie algebra of a compact Lie group is negative semidefinite, not negative definite in general.
If the Killing form of a Lie algebra is negative definite, then the Lie algebra is the Lie algebra of a compact semisimple Lie group.
In general, the Lie algebra of a compact Lie group decomposes as the Lie algebra direct sum of a commutative summand (for which the corresponding subgroup is a torus) and a summand on which the Killing form is negative definite.
It is important to note that the converse of the first result above is false: Even if the Killing form of a Lie algebra is negative semidefinite, this does not mean that the Lie algebra is the Lie algebra of some compact group. For example, the Killing form on the Lie algebra of the Heisenberg group is identically zero, hence negative semidefinite, but this Lie algebra is not the Lie algebra of any compact group.
Compact Lie algebras are reductive; note that the analogous result is true for compact groups in general.
The Lie algebra for the compact Lie group G admits an Ad(G)-invariant inner product,. Conversely, if admits an Ad-invariant inner product, then is the Lie algebra of some compact group. If is semisimple, this inner product can be taken to be the negative of the Killing form.
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