Compact Lie algebra
= Compact Lie algebra
A real Lie algebra is compact when it is the Lie algebra of some <compact Lie group>. Equivalently, it has a positive-definite <invariant bilinear form on a Lie algebra>. It is a direct sum of an abelian center and a compact <semisimple Lie algebra>. For its semisimple part, the <compactness criterion from the Killing form> is negative-definiteness of the <Killing form>. This definition concerns the algebra; a global group with that algebra can still have a noncompact central covering direction.