2.9. Lie algebras 25

they were called “infinitesimal groups” and were called Lie algebras

by Hermann Weyl in honor of Lie). A Lie group is a group G which is

also a manifold (i.e., a topological space which locally looks like

Rn)

such that the multiplication operation is differentiable. In this case,

one can define the algebra of smooth functions

C∞(G)

which carries

an action of G by right translations ((g ◦ f)(x) := f(xg)), and the

Lie algebra Lie(G) of G consists of derivations of this algebra which

are invariant under this action (with the Lie bracket being the usual

commutator of derivations). Clearly, such a derivation is determined

by its action at the unit element e ∈ G, so Lie(G) can be identified

as a vector space with the tangent space TeG to G at e.

Sophus Lie showed that the attachment G → Lie(G) is a bijec-

tion between isomorphism classes of simply connected Lie groups (i.e.,

connected Lie groups on which every loop contracts to a point) and

finite dimensional Lie algebras over R. This allows one to study (dif-

ferentiable) representations of Lie groups by studying representations

of their Lie algebras, which is easier since Lie algebras are “linear” ob-

jects while Lie groups are “nonlinear”. Namely, a finite dimensional

representation of G can be differentiated at e to yield a representa-

tion of Lie(G), and conversely, a finite dimensional representation of

Lie(G) can be exponentiated to give a representation of G. Moreover,

this correspondence extends to certain classes of infinite dimensional

representations.

The most important examples of Lie groups are linear algebraic

groups, which are subgroups of GLn(R) defined by algebraic equa-

tions (such as, for example, the group of orthogonal matrices On(R)).

Also, given a Lie subalgebra g ⊂ gln(R) (which, by Ado’s theo-

rem, can be any finite dimensional real Lie algebra), we can define G

to be the subgroup of GLn(R) generated by the elements

eX

, X ∈ g.

One can show that this group has a natural structure of a connected

Lie group, whose Lie algebra is g (even though it is not always a

closed subgroup). While this group is not always simply connected,

its universal covering G is, and it is the Lie group corresponding to g

under Lie’s correspondence.

For more on Lie groups and their relation to Lie algebras, the

reader is referred to textbooks on this subject, e.g. [Ki].