2 1. INTRODUCTION
term of the exponent in the Campbell-Baker-Hausdorff formula
exp(X)exp(Y ) = exp(X + Y +
(XY − Y X) + · · · ),
we can recover information on the group structure of the Lie group. (This is no
longer true if we treat matrix groups over fields of positive characteristic. Here, we
consider Lie groups over C only.) Since we do not treat Lie groups in these lectures,
we start with Lie algebras.
Definition 1.1. Let g be a vector space over C equipped with a C-bilinear
map, called the Lie bracket, [ , ] : g ⊗ g → g satisfying:
(1) [Y, X] + [X, Y ] = 0,
(2) [X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y ]] = 0.
Then g is called a Lie algebra over C. The condition (2) is called the Jacobi
A linear map φ between two Lie algebras is called a Lie algebra homomor-
phism if φ([X, Y ]) = [φ(X), φ(Y )] for X, Y ∈ g.
Let V be a vector space over C. Then End(V ) becomes a Lie algebra via
[X, Y ] := XY −Y X. We denote this Lie algebra by gl(V ). If a basis of V is chosen
and V is identified with
it is also denoted by gl(n, C). The following definition
is also important.
Definition 1.2. A subspace a of a Lie algebra g is called a Lie subalgebra
of g if the condition [a, a] ⊂ a is satisfied. If [a, a] = 0 holds, a is always a Lie
subalgebra. In this case, a is called a commutative Lie subalgebra.
If a satisfies a stronger condition [g, a] ⊂ a, we call a an ideal of g.
The notion of representations for Lie algebras is defined in the following way.
Definition 1.3. Let V be a (not necessarily finite dimensional) vector space
over C. If a C-linear map ρ : g → End(V ) satisfies
ρ([X, Y ]) = ρ(X)ρ(Y ) − ρ(Y )ρ(X),
then ρ is called a representation of g , and V is called a g-module.
In other words, if ρ : g → gl(V ) is a Lie algebra homomorphism, ρ is called a
representation of g. If one would like to record both ρ and V explicitly, we make
these into a pair (ρ, V ), and call it a representation.
An important example is the adjoint representation (ad, g), where ad : g →
End(g) is defined by ad(X)(Y ) = [X, Y ].
Although Lie algebras appear to be strange algebras, they are in fact connected
with usual (associative) algebras, and we can interpret results about a Lie algebra
as results about an algebra in the usual sense; this algebra is the enveloping
algebra of g, whose precise definition is given later. In Lie theory, it has become
more popular to work with enveloping algebras.
Another important work relevant to us is a result of Serre. His result is that
we can give a definition of enveloping algebras in terms of generators and relations
for a good class of Lie algebras, semisimple Lie algebras. These relations are now
called the Serre relations.
As research developed this way, an interesting breakthrough was made in the
1980’s by leading mathematicians in the Kyoto school. They discovered that if we