Compact Lie algebra

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;[1] 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.

[2] 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:[2] 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.

The compact Lie algebras are classified and named according to the compact real forms of the complex semisimple Lie algebras.

These are: The classification is non-redundant if one takes

one obtains certain exceptional isomorphisms.

is the trivial diagram, corresponding to the trivial group

{\displaystyle \operatorname {SU} (1)\cong \operatorname {SO} (1)\cong \operatorname {Sp} (0)\cong \operatorname {SO} (0).}

{\displaystyle {\mathfrak {su}}_{2}\cong {\mathfrak {so}}_{3}\cong {\mathfrak {sp}}_{1}}

corresponds to the isomorphisms of diagrams

and the corresponding isomorphisms of Lie groups

⁡ ( 2 ) ≅ Spin ⁡ ( 3 ) ≅ Sp ⁡ ( 1 )

{\displaystyle \operatorname {SU} (2)\cong \operatorname {Spin} (3)\cong \operatorname {Sp} (1)}

(the 3-sphere or unit quaternions).

corresponds to the isomorphisms of diagrams

and the corresponding isomorphism of Lie groups

Sp ⁡ ( 2 ) ≅ Spin ⁡ ( 5 ) .

corresponds to the isomorphisms of diagrams

and the corresponding isomorphism of Lie groups

{\displaystyle \operatorname {SU} (4)\cong \operatorname {Spin} (6).}

respectively, with corresponding isomorphisms of Lie algebras.