Springer GTM 9.

Representation theory, Lie Theory

# Chapter I: Basic Concepts Edit

## Section I.1: Definitions and first examples Edit

## Section I.2: Ideals and homomorphisms Edit

### Subsection I.2.1: Ideals Edit

- The basic terminology here is confusing. Some of the terminology is coming from the fact that Lie algebras are algebras over fields, albeit strange non-associative ones, so for instance an ideal of a Lie algebra is an exactly what it is for any other algebra or ring: we think of the Lie bracket as ring multiplication. On the other hand, some of the terminology comes from the fact that Lie algebras sit below Lie groups. For instance, a Lie algebra is abelian if the corresponding Lie group is, which means that we think of the Lie bracket in this case as corresponding to a commutator in the Lie group.

- Looking at the point above, normal subgroups, and by extension normalizers, happen in groups and not rings, so when we talk about the normalizer of a subspace we can safely assume that we're regarding the bracket as a commutator. We have to translate the usual notion of normalizers into the language of commutators, though: For a subgroup H of a group G, $ N_G(H) = \{ x \in G \; | \; x h x^{-1} \in H \;\forall h \in H \} $. But $ x h x^{-1} \in H $ if and only if $ [x, h] = x h x^{-1} h^{-1} \in H h^{-1} = H $, so we can re-phrase this definition as $ N_G(H) = \{ x \in G \; | \; [x, H] \subseteq H\} $.

Group-theoretic / Commutator | Ring-theoretic / Ring product |
---|---|

center / abelian | ideal |

derived algebra | quotient algebra |

normalizer, centralizer | nilpotent |

simple | |

solvable |

For $ A, B \subseteq L $, let $ (A : B) = \{ x \in L : [x, B] \subseteq A \} $. (This is variously known as the transporter, conductor, or ideal quotient, and may require some modifications to deal with contexts which aren't at least roughly commutative.) Then we can translate some of these terms into more ring-theoretic language.

Lie algebra terminology | Ring-theoretic terminology |
---|---|

Z(L) | (0:L) = Ann L |

C(X) | (0:X) = Ann X |

N(K) | (K:K) = I(K) "idealizer" |

abelian | $ L^2 = 0 $ |

derived algebra | $ L^2 $ |

### Subsection I.2.2: Homomorphisms and representations Edit

### Subsection I.2.3: Automorphisms Edit

- I'm pretty sure that this is a translation of something that would make a lot more sense in terms of Lie groups, but I haven't actually worked out what's going on yet. (I suspect we're just looking at what conjugation in a Lie group does to the Lie algebra.)

## Section I.3: Solvable and Nilpotent Lie algebras Edit

### Subsection I.3.1: Solvability Edit

### Subsection I.3.2: Nilpotence Edit

- L is nilpotent if there is some n for which, for every $ x_1, x_2, \ldots, x_n \in L $, we have $ [x_1, [x_2, [x_3, \ldots x_n]\ldots]]=0 $.
- An element $ x \in L $, on the other hand, is ad-nilpotent if $ [x, [x, [x, \ldots y]\ldots]] = 0 $ for every $ y \in L $.
- These are two different notions of nilpotence. A Lie algebra L is nilpotent if $ L^n = 0 $; here, we're viewing the Lie bracket as a ring product and applying the usual definition of nilpotence in rings. An element of a Lie algebra is ad-nilpotent, on the other hand, if (ad x), viewed as an element of the (associative) endomorphism ring End L, is nilpotent.
- The statement of the lemma should really have End V instead of gl(V) since we're viewing x as an endomorphism, nor an element of a Lie algebra.

### Subsection I.3.3: Proof of Engel's Theorem Edit

- The statement that, if K is a subalgebra of a Lie algebra L, then ad K acts on the vector space L/K, doesn't depend on the hypothesis and should probably be factored out of the proof of Theorem 3.3 and made into a lemma.