Orthogonal complements of semisimple ideals

There is a general fact about semisimple ideals in arbitrary Lie algebras that we prove next; it will be an application of the material on complete reducibility.  We will use it to complete the picture of the abstract Jordan decomposition in a semisimple Lie algebra.

Proposition 1 Let ${\mathfrak{s} \subset \mathfrak{g}}$ be a semisimple ideal in the Lie algebra ${\mathfrak{g}}$. Then there is a unique ideal ${\mathfrak{a} \subset \mathfrak{g}}$ with ${\mathfrak{g} = \mathfrak{s} \oplus \mathfrak{a}}$.

The idea is that we can find a complementary ${\mathfrak{s}}$-module ${A \subset \mathfrak{g}}$ with

$\displaystyle \mathfrak{g} = \mathfrak{s} \oplus A$

as ${\mathfrak{s}}$-modules. Now ${[\mathfrak{s},A] \subset A \cap \mathfrak{s} = \{0\}}$ because ${\mathfrak{s}}$ is an ideal and because ${A}$ is stable under the action of ${\mathfrak{s}}$. The converse is true: ${A}$ is the centralizer of ${\mathfrak{s}}$, and thus an ideal, because anything commuting with ${\mathfrak{s}}$ can have no part in ${\mathfrak{s}}$ in the ${\mathfrak{s} \oplus A}$ decomposition (${\mathfrak{s}}$ being semisimple). Thus, I hereby anoint ${A}$ with the fraktur font and call it ${\mathfrak{a}}$ to recognize its Lie algebraness.

Given a splitting as above, ${\mathfrak{a}}$ would have to be the centralizer of ${\mathfrak{s}}$, so uniqueness is evident. (more…)

For the next few weeks, I’m probably going to be doing primarily algebra posts.

Invariant bilinear forms

Let ${\mathfrak{g}}$ be a Lie algebra over the field ${k}$ and ${V,W}$ representations. Recall the following.

1. ${v \in V}$ is invariant under ${\mathfrak{g}}$ if ${Xv = 0}$ for all ${X \in \mathfrak{g}}$.

2. ${\hom_k(V,W)}$ is a representation of ${\mathfrak{g}}$: define $(Xf)v = X(fv) - f(Xv)$. This is isomorphic as a ${\mathfrak{g}}$-module to the tensor product ${W \otimes V^{\vee}}$, where ${V^{\vee}}$ is regarded as a ${\mathfrak{g}}$-module. We can think of ${W \otimes V^{\vee}}$ as a ${\mathfrak{g}}$-module because the enveloping algebra ${U\mathfrak{g}}$ is a Hopf algebra under the homomorphism ${U\mathfrak{g} \rightarrow U \mathfrak{g} \otimes U \mathfrak{g}}$ given by ${x \mapsto 1 \otimes x + x \otimes 1}$ for ${x \in \mathfrak{g}}$, and extended further.

3. Let ${B}$ be a bilinear form on ${V}$, i.e. a linear map ${B: V \otimes V \rightarrow k}$. Then ${B}$ is said to be invariant under ${\mathfrak{g}}$ if for all ${v,v' \in V, X \in \mathfrak{g}}$

$\displaystyle B(Xv, v') + B(v, Xv') = 0;$

if we treat ${B}$ as an element of ${(V \otimes V)^{\vee}}$, this is the same as saying it is invariant in the sense of 1 above.

Ok, all good. Given a representation ${V}$ as above, we have a particular example ${B_V}$ of an invariant and symmetric bilinear form on ${\mathfrak{g}}$ in this post (which in particular shows why some of what I just posted here is redundant; I hadn’t looked back when I started writing it) given by

$\displaystyle B_V(x,y) := \mathrm{Tr}( x_V y_V),$

where ${x_V,y_V}$ are the corresponding endomorphisms of ${V}$ corresponding to ${x,y \in \mathfrak{g}}$. An important special case of this is when we are considering the adjoint representation of ${\mathfrak{g}}$ on itself; then this is called the Killing form.

What we shall prove is the following:

Theorem 1 (Cartan)

Let the ground field ${k}$ be of characteristic zero. The Lie algebra ${\mathfrak{g}}$ is solvable if and only if$\displaystyle B(\mathfrak{g}, [\mathfrak{g},\mathfrak{g}]) = \{ 0 \},$

for ${B}$ the Killing form of the adjoint representation. (more…)

${\mathfrak{sl}_2}$ 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.