# Wednesday, September 14

## 6.1: Structure theory

:::{.definition title="Natural representation"}
Note that all of the algebras $\lieg$ we've considered naturally act on column vectors in some $\FF^n$ -- this is the **natural representation** of $\lieg$.
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:::{.example title="?"}
Letting $\lieb_n$ be the upper triangular matrices in $\liegl_n$, this acts on $\FF^n$.
Taking a standard basis $\FF^n = V \da \gens{e_1,\cdots, e_n}_\FF$, one gets submodules $V_i = \gens{e_1,\cdots, e_i}_\FF$ which correspond to upper triangular blocks got by truncating the first $i$ columns of the matrix.
This yields a submodule precisely because the lower-left block is zero.
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:::{.remark}
Let $\phi: L\to \liegl(V)$ be a representation, noting that $\Endo(V)$ is an associative algebra.
We can consider the associative unital algebra $A$ generated by the image $\phi(L)$.
Note that the structure of $V$ as an $L\dash$module is the same as its $A\dash$module structure, so we can apply theorems/results from the representation of rings and algebras to study Lie algebra representations, e.g. the Jordan-Holder theorem and Schur's lemma.
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:::{.definition title="Direct sums of $L\dash$modules"}
Given $V, W\in \mods{L}$, their vector space direct sum admits an $L\dash$module structure using $x.(v, w) \da (x.v, x.w)$, which we'll write as $x.(v+w) \da xv + xw$.
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:::{.definition title="Completely reducible modules"}
$V\in \mods{L}$ is **completely reducible** iff $V$ is a direct sum of irreducible $L\dash$modules.
Equivalently, for each $W\leq V$ there is a complementary submodule $W'$ such that $V = W \oplus W'$.
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:::{.warnings}
"Not irreducible" is strictly weaker than "completely reducible", since a submodule may not admit an invariant complement -- for example, the flag in the first example above.
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:::{.example title="?"}
The natural representation of $\lieh_n$ is completely reducible, decomposing as $V_1 \oplus V_2 \oplus \cdots V_n$ where $V_i = \FF e_i$.

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:::{.definition title="Indecomposable modules"}
A module $V$ is **indecomposable** iff $V\neq W \oplus W'$ for proper submodules $W, W' \leq V$.
This is weaker than irreducibility.
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:::{.example title="?"}
Consider the natural representation $V$ for $L \da \lieb_n$.
Every nonzero submodule of $V$ must contain $e_1$, so $V$ is indecomposable if $n\geq 1$.
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:::{.remark}
Recall that the **socle** of $V$ is the (direct) sum of all of its irreducible submodules.
If $\soc(V)$ is simple (so one irreducible) then $V$ is indecomposable, since every summand must contain this simple submodule "at the bottom".
For $L = \lieb_n$, note that $\soc(V) = \FF e_1$.
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:::{.remark}
For the reminder of chapters 6 and 7, we assume all modules are finite-dimensional over $\FF = \bar\FF$.
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:::{.theorem title="Jordan-Holder"}
Let $L\in\Lie\Alg\slice\FF^\fd$, then there exists a **composition series**, a sequence of submodules $0 = V_0 \subseteq V_1 \subseteq \cdots \subseteq V_n = V$ such that each composition factor (sometimes called *sections*) $V_i/V_{i-1}$ is irreducible/simple.
Moreover, any two composition series admit the same composition factors with the same multiplicities, up to rearrangement and isomorphism.
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:::{.example title="?"}
If $V = W \oplus W'$ with $W, W'$ simple, there are two composition series:

- $0 \subseteq W \subseteq V$ with factors $W, V/W \cong W'$,
- $0 \subseteq W' \subseteq V$ with factors $W', V/W' \cong W$.

These aren't equal, since they're representations on different coset spaces, but are isomorphic.
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:::{.theorem title="Schur's lemma"}
If $\phi: L\to \liegl(V)$ is an irreducible representation, then $\Endo_L(V)\cong \FF$.
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:::{.proof title="?"}
If $V$ is irreducible then every $f\in \mods{L}(V, V)$ is either zero or an isomorphism since $f(V) \leq V$ is a submodule.
Thus $\Endo_L(V)$ is a division algebra over $\FF$, but the only such algebra is $\FF$ since $\FF = \bar\FF$.

Letting $\phi$ be as above, it has an eigenvalue $\lambda \in \FF$, again since $\FF = \bar\FF$.
Then $\phi - \lambda I \in \Endo_L(V)$ has a nontrivial kernel, the $\lambda\dash$eigenspace.
So $\phi - \lambda I = 0 \implies \varphi = \lambda I$.
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:::{.warnings}
Schur's lemma is not always true for Lie *superalgebras*.
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:::{.definition title="Trivial modules"}
The **trivial $L\dash$module** is $\FF \in \mods{L}$ equipped with the zero map \( \varphi: L\to \liegl(\FF) \) where $x.1 \da 0$ for all $x\in L$.
Note that this is irreducible, and any two such 1-dimensional trivial modules are isomorphic by sending a basis $\ts{e_1}$ to $1\in \FF$.

More generally, an $V \in \mods{L}$ is trivial iff $x.v = 0$ for all $x\in L, v\in V$, and $V$ is completely reducible as a direct sum of copies of the above trivial module.
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:::{.definition title="Homs, Tensors, and duals"}
Let $V, W\in \mods{L}$, then the following are all $L\dash$modules:

- $V\tensor_\FF W$: the action is $x.(v \tensor w) = (x.v)\tensor w + v\tensor(x.w)$.[^deriv_group]
- $\Hom_\FF(V, W)$: the action is $(x.f)(v) = x.(f(v)) - f(x.v) \in W$.
- $V\dual \da \Hom_\FF(V, \FF)$: the action is a special case of the above since $x.w = 0$, so[^inverse_dif]
\[
(x.f)(v) = -f(x.v)
.\]

[^inverse_dif]: 
One might expect an inverse from group theory, which differentiates to a minus sign.

[^deriv_group]: 
Note that groups would act on each factor separately, and this is more like a derivative.

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:::{.remark}
These structures come from the Hopf algebra structure on the universal associative algebra $U(\lieg)$, called the **universal enveloping algebra**.
Note that we also have
\[
\Hom_\CC(V, W)\isoas{\mods{L}} V\dual \tensor_\FF W
.\]

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