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].
Previous Page Next Page