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More Lie algebra constructions July 28, 2009

Posted by Akhil Mathew in algebra, representation theory.
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The ultimate aim in the series on Lie algebras I am posting here is to cover the representation theory of semisimple Lie algebras. To get there, we first need to discuss some technical tools—for instance, invariant bilinear forms.

Generalities on representations

Fix a Lie algebra {L}. Given representations {V_1, V_2}, we clearly have a representation {V_1 \oplus V_2}; given a morphism of representations {V_1 \rightarrow V_2}, i.e. one which respects the action of {L}, the kernel and image are themselves representations.

Proposition 1 The category {Rep(L)} of finite-dimensional representations of {L} is an abelian category.

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Lie’s Theorem II July 27, 2009

Posted by Akhil Mathew in algebra, representation theory.
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Yesterday I was talking about Lie’s theorem for solvable Lie algebras. I went through most of the proof, but didn’t finish the last step. We had a solvable Lie algebra {L} and an ideal {I \subset L} such that {I} was of codimension one.

There was a finite-dimensional representation {V} of {L}. For {\lambda \in I^*}, we set

\displaystyle  V_\lambda := \{ v \in V: Yv = \lambda(Y) v, \ \mathrm{all} \ Y \in I \}.

We assumed {V_\lambda \neq 0} for some {\lambda} by the induction hypothesis. Then the following then completes the proof of Lie’s theorem, by the “fundamental calculation:”

Lemma 1 If {V_\lambda \neq 0}, then {\lambda([L,I])=0}.

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Lie’s Theorem I July 26, 2009

Posted by Akhil Mathew in algebra, representation theory.
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I talked a bit earlier about nilpotent Lie algebras and Engel’s theorem. There is an analog for solvable Lie algebras, and the corresponding Lie’s theorem.

So, first the definitions. Solvability is similar to nilpotence in that one takes repeated commutators, except one uses the derived series instead of the lower central series.

In the future, fix a Lie algebra {L} over an algebraically closed field {k} of characteristic zero.

Definition 1 The derived series of {L} is the descending filtration {D_n} defined by {D_0 := L, D_n := [D_{n-1}, D_{n-1}]}. The Lie algebra {L} is solvable if {D_M=0} for some {M}.

For instance, a nilpotent Lie algebra is solvable, since if {\{C_n\}} is the lower central series, then {D_n \subset C_n} for each {n}.

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The enveloping algebra July 25, 2009

Posted by Akhil Mathew in algebra.
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As we saw in the first post, a representation of a finite group {G} can be thought of simply as a module over a certain ring: the group ring. The analog for Lie algebras is the enveloping algebra. That’s the topic of this post.

Definition

The basic idea is as follows. Just as a representation of a finite group {G} was a group-homomorphism {G \rightarrow Aut(V)} for a vector space, a representation of a Lie algebra {\mathfrak{g}} is a Lie-algebra homomorphism {\mathfrak{g} \rightarrow \mathfrak{g}l(V)}. Now, {\mathfrak{g}l(V)} is the Lie algebra constructed from an associative algebra, {End(V)}—just as {Aut(V)} is the group constructed from {End(V)} taking invertible elements.

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Engel’s Theorem and Nilpotent Lie Algebras July 23, 2009

Posted by Akhil Mathew in algebra, representation theory.
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Now that I’ve discussed some of the basic definitions in the theory of Lie algebras, it’s time to look at specific subclasses: nilpotent, solvable, and eventually semisimple Lie algebras. Today, I want to focus on nilpotence and its applications.

Engel’s Theorem

To start with, choose a Lie algebra {L \subset \mathfrak{gl} (V)} for some finite-dimensional {k}-vector space {V}; recall that {\mathfrak{gl} (V)} is the Lie algebra of linear transformations {V \rightarrow V} with the bracket {[A,B] := AB - BA}. The previous definition was in terms of matrices, but here it is more natural to think in terms of linear transformations without initially fixing a basis.

Engel’s theorem is somewhat similar in its statement to the fact that commuting diagonalizable operators can be simultaneously diagonalized.

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Lie algebras II July 20, 2009

Posted by Akhil Mathew in algebra.
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I’m going to get back eventually to the story about finite-dimensional modules, but for now, Lie algebras are more immediate to my project, so I’ll talk about them here.

From an expository standpoint, jumping straight to {\mathfrak{sl}_2} basically right after defining Lie algebras was unsound. I am going to try to motivate them here and discuss some theorems, to lead into more of the general representation theory.

Derivations

So let’s consider a not-necessarily-associative algebra {A} over some field {F}. In other words, {A} is a {F}-vector space, and there is a {F}-bilinear map {A \times A \rightarrow A}, which sends say {(x,y) \rightarrow xy}, but it doesn’t have to either be commutative or associative (or unital). A Lie algebra with the Lie bracket would be one example.

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Representations of sl2, Part II July 18, 2009

Posted by Akhil Mathew in algebra, representation theory.
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This post is the second in the series on {\mathfrak{sl}_2} and the third in the series on Lie algebras. I’m going to start where we left off yesterday on {\mathfrak{sl}_2}, and go straight from there to classification.  Basically, it’s linear algebra.

Classification

We’ve covered all the preliminaries now and we can classify the {\mathfrak{sl}_2}-representations, the really interesting material here. By Weyl’s theorem, we can restrict ourselves to irreducible representations. Fix an irreducible {V}.

So, we know that {H} acts diagonalizably on {V}, which means we can write

\displaystyle  V = \bigoplus_\lambda V_\lambda

where {Hv_\lambda = \lambda v_{\lambda}} for each {\lambda}, i.e. {V_\lambda} is the {H}-eigenspace.

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Representations of sl2, Part I July 17, 2009

Posted by Akhil Mathew in algebra, representation theory.
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{\mathfrak{sl}_2} is a special Lie algebra, mentioned in my previous post briefly. It is the set of 2-by-2 matrices over {\mathbb{C}} of trace zero, with the Lie bracket defined by:

\displaystyle  [A,B] = AB - BA.

The representation theory of {\mathfrak{sl}_2} is important for several reasons.

  1. It’s elegant.
  2. It introduces important ideas that generalize to the setting of semisimple Lie algebras.
  3. Knowing the theory for {\mathfrak{sl}_2} is useful in the proofs of the general theory, as it is often used as a tool there.

In this way, {\mathfrak{sl}_2} is an ideal example. Thus, I am posting this partially to help myself learn about Lie algebras.

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Lie algebras: fundamentals July 16, 2009

Posted by Akhil Mathew in algebra, representation theory, Uncategorized.
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The following topic came up in the context of my project, which has been expanding to include new areas of mathematics that I did not initially anticipate. Consequently, I have had to learn about several new areas of mathematics; this is, of course, a common experience at RSI. For me, the representation theory of Lie algebras has been one of those areas, and I will post here about it to help myself understand it. Right here, I’ll aim to cover the groundwork necessary to get to the actual representation theory in a future post.

Lie Algebras

Throughout, we work over {{\mathbb C}}, or even an algebraically closed field of characteristic zero.

Definition 1 A Lie algebra is a finite-dimensional vector space {L} with a Lie bracket {[\cdot, \cdot]: L \times L \rightarrow L} satisfying:

  • The bracket {[\cdot, \cdot]: L \times L \rightarrow L} is {{\mathbb C}}-bilinear in both variables.
  • {[A,B] = -[B,A]} for any {A,B \in L}.
  • {[A, [B,C]] + [B, [C,A]] + [C, [A,B]] = 0}. This is the Jacobi identity.

To elucidate the meaning of the conditions, let’s look at a few examples. (more…)

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