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!).
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