In the mathematical field of Lie theory, there are [|two definitions] of a compactLie 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.
Definition
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 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.
The Lie algebra for the compact Lie group G admits an Ad-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. Thus relative to this inner product, Ad acts by orthogonal transformations and acts by skew-symmetric matrices. It is possible to develop the theory of complex semisimple Lie algebras by viewing them as the complexifications of Lie algebras of compact groups; the existence of an Ad-invariant inner product on the compact real form greatly simplifies the development.
:This can be seen as a compact analog of Ado's theorem on the representability of Lie algebras: just as every finite-dimensional Lie algebra in characteristic 0 embeds in every compact Lie algebra embeds in
Compact real forms of the exceptional Lie algebras
Isomorphisms
The classification is non-redundant if one takes for for for and for If one instead takes or one obtains certain exceptional isomorphisms. For is the trivial diagram, corresponding to the trivial group For the isomorphism corresponds to the isomorphisms of diagrams and the corresponding isomorphisms of Lie groups . For the isomorphism corresponds to the isomorphisms of diagrams and the corresponding isomorphism of Lie groups For the isomorphism corresponds to the isomorphisms of diagrams and the corresponding isomorphism of Lie groups If one considers and as diagrams, these are isomorphic to and respectively, with corresponding isomorphisms of Lie algebras.
