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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