Compact Lie algebras and Lie groups

Question

A simple or semisimple Lie algebra is said to be compact if the $\mathrm{Tr}\left \{ T^\mathrm{adj}_{a}, T^\mathrm{adj}_{b}\right \}$ is positive definite where $T^\mathrm{Adj}_{a}$ are the generators of the adjoint representation. How does compact Lie algebras defined in this way are related to compact Lie groups? How can I prove that they are the Lie algebras of compact Lie groups?

This post imported from StackExchange Mathematics at 2014-06-14 08:29 (UCT), posted by SE-user Bill

Comments

I am not sure compact lie algebras and lie groups should be related, because locally compact and non-compact manifolds look the same.

This post imported from StackExchange Mathematics at 2014-06-14 08:29 (UCT), posted by SE-user ramanujan_dirac

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From wikipedia (en.wikipedia.org/wiki/Compact_Lie_algebra) it seems I was wrong, but wikipedia defines the killing form to be negative definite.

This post imported from StackExchange Mathematics at 2014-06-14 08:29 (UCT), posted by SE-user ramanujan_dirac

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I think this question is basically answered by the Wikipedia article (provided that you change the definition to "negative definite"). You need to exclude tori to get a precise correspondence.

This post imported from StackExchange Mathematics at 2014-06-14 08:29 (UCT), posted by SE-user Qiaochu Yuan

Answer 1

to add to SDevalapurkar's answer:

Another good reference for the relationship between a compact (in the topological sense) Lie group and one whose Lie algebra is compact is S. Helgason "Differential geometry Lie groups and symmetric spaces" 

Although you have to make the (small IMO) further assumption that the group has finite centre, you then have the following definitive characterisation:

Proposition: Given that a Lie group has a finite centre, then the Killing form on a group's Lie algebra is negative definite if and only if the group is compact (in the topological sense).

(i.e. we consider only the class of Lie groups with finite centres, not all Lie groups)

The proof is in Chap. II, section 6, prop. 6.6 of the Helgason reference I cite.

Answer 2

A compact Lie algebra can be defined in either of the following two ways:

1. The Lie algebra of a compact Lie group.
2. A real Lie algebra whose Killing form is negative-definite.

For 1., the Lie algebra of a compact Lie group has a negative semidefinite Killing form (hence it's a compact reductive Lie algebra) by Knapp, Lie Groups Beyond an Introduction, Proposition 4.26; thus every compact Lie algebra (hence it's a compact semisimple Lie algebra) defined by 2. is a compact Lie algebra as defined by 1.