22 2. Basic notions of representation theory

2.9. Lie algebras

Let g be a vector space over a field k, and let [ , ] : g × g −→ g

be a skew-symmetric bilinear map. (That is, [a, a] = 0, and hence

[a, b] = −[b, a].)

Definition 2.9.1. (g, [ , ]) is a Lie algebra if [ , ] satisfies the Jacobi

identity

(2.9.1) [a, b] , c + [b, c] , a + [c, a] , b = 0.

Example 2.9.2. Some examples of Lie algebras are:

(1) Any space g with [ , ] = 0 (abelian Lie algebra).

(2) Any associative algebra A with [a, b] = ab−ba , in particular,

the endomorphism algebra A = End(V ), where V is a vector

space. When such an A is regarded as a Lie algebra, it is

often denoted by gl(V ) (general linear Lie algebra).

(3) Any subspace U of an associative algebra A such that [a, b] ∈

U for all a, b ∈ U.

(4) The space Der(A) of derivations of an algebra A, i.e. linear

maps D : A → A which satisfy the Leibniz rule:

D(ab) = D(a)b + aD(b).

(5) Any subspace a of a Lie algebra g which is closed under the

commutator map [ , ], i.e., such that [a, b] ∈ a if a, b ∈ a.

Such a subspace is called a Lie subalgebra of g.

Remark 2.9.3. Ado’s theorem says that any finite dimensional Lie

algebra is a Lie subalgebra of gl(V ) for a suitable finite dimensional

vector space V .

Remark 2.9.4. Derivations are important because they are the “in-

finitesimal version” of automorphisms (i.e., isomorphisms onto itself).

For example, assume that g(t) is a differentiable family of automor-

phisms of a finite dimensional algebra A over R or C parametrized

by t ∈ (−, ) such that g(0) = Id. Then D := g (0) : A → A is a

derivation (check it!). Conversely, if D : A → A is a derivation, then

etD

is a 1-parameter family of automorphisms (give a proof!).