Initial Classification of Lie Algebras
In this lecture we define various subclasses of Lie algebras: nilpotent, solvable, semi-simple, etc., and prove basic facts about their representations. The discussion is entirely elementary (largely because the hard theorems are stated without proof for
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Initial Classification of Lie Algebras
In this lecture we define various subclasses of Lie algebras: nilpotent, solvable, semisimple, etc., and prove basic facts about their representations. The discussion is entirely elementary (largely because the hard theorems are stated without prooffor now); there are no prerequisites beyond linear algebra. Apart from giving these basic definitions, the purpose of the lecture is largely to motivate the narrowing of our focus to semisimple algebras that will take place in the sequel. In particular, the first part of §9.3 is logically the most important for what follows. §9.1: §9.2: §9.3: §9.4:
Rough classification of Lie algebras Engel's Theorem and Lie's Theorem Semisimple Lie algebras Simple Lie algebras
§9.1. Rough Classification of Lie Algebras We will give, in this section, a preliminary sort of classification of Lie algebras, reflecting the degree to which a given Lie algebra 9 fails to be abelian. As we have indicated, the goal ultimately is to narrow our focus onto semisimpie Lie algebras. Before we begin, two definitions, both completely straightforward: First, we define the center Z(g) of a Lie algebra 9 to be the subspace of 9 of elements X E 9 such that [X, Y] = 0 for all Y E g. Of course, we say 9 is abelian if all brackets are zero. Exercise 9.1. Let G be a Lie group, 9 its Lie algebra. Show that the subgroup of G generated by exponentiating the Lie subalgebra Z(g) is the connected component of the identity in the center Z(G) of G. W. Fulton et al., Representation Theory © Springer Science+Business Media, Inc. 2004
9. Initial Classification of Lie Algebras
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Next, we say that a Lie subalgebra 1) c 9 of a Lie algebra 9 is an ideal if it satisfies the condition [X, y] E 1)
for all X E 1), Y E g.
Just as connected subgroups of a Lie group correspond to subalgebras of its Lie algebra, the notion of ideal in a Lie algebra corresponds to the notion of normal subgroup, in the following sense: Exercise 9.2. Let G be a connected Lie group, H eGa connected subgroup and 9 and 1) their Lie algebras. Show that H is a normal subgroup of G if and only if 1) is an ideal of g. Observe also that the bracket operation on 9 induces a bracket on the quotient space g/1) if and only if 1) is an ideal in g. This, in turns, motivates the next bit of terminology: we say that a Lie algebra 9 is simple if dim 9 > 1 and it contains no nontrivial ideals. By the last exercise, this is equivalent to saying that the adjoint form G of the Lie algebra 9 has no nontrivial normal Lie subgroups. Now, to attempt to classify Lie algebras, we introduce two descending chains of subalgebras. The first is the lower central series of subalgebras ~kg, defined inductively by ~lg
= [g, g]
and ~kg =
[g, ~k-lg].
Note that the subalgebras ~kg are in fact ideals in g. The other series is called the derived series {~kg}; it is defined by ~lg =
[g, g]
and
Exercise 9.3. Use the Jacobi identity to show that ~kg is also an ideal in g. More generally, if 1) is an ideal in a Lie algebra g, show that [1),
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