# Monday October 21

## Chapter 5: Existence Theorem

### Universal Enveloping Algebra (UAE)

Some applications/motivations for UAEs:

1. Groups $G$ are to group algebras $F[G]$ as Lie algebras $\lieg$ are to UAE $U(\lieg)$.
  Any $\lieg\dash$module then becomes a module over a ring, so the general theory applies.
2. PBW theorem: this yields a concrete $F\dash$basis of $U(\lieg)$. 
  There is a triangular decomposition $U(\lieg) = U(h) \tensor U(f) \tensor U(n)$.
  This allows constructing the Vermo module (and hence irreducible modules) for $\lieg$,
  allowing for a description of BGG Category $\mathcal O$.
3. Harish-Chandra theorem: $Z(U(\lieg)) = S(\lieg)^W$.
  This characterizes central characters $\chi: Z(U(\lieg)) \to F$,
  which further allows describing the blocks of $\mathcal O$, i.e. when two irreducible modules have non-trivial extensions.
4. $U(\lieg)$ deforms to a quantum group $U_q(\lieg)$.

### Tensor and Symmetric Algebras

**Definition:**
For $V$ a f.d. vector space, the *tensor algebra* over $V$ is given by $T(V) = \bigoplus_{n\in\NN} T^n(V)$ where
$T^n(V) = \tensor_{i=1}^n V$ with an associative multiplication $T^a \cross T^b \to T^{a+b}$ given by $(\tensor_{i=1}^a v_i, \tensor_{i=1}^b w_i) \mapsto \tensor_{i=1}^a v_i \tensor \tensor_{i=1}^b w_i$.

The tensor algebra satisfies a universal property: given any $F\dash$linear map $\phi: V \to A$. (See phone image)

**Definition:**
The symmetric algebra on $V$ is given by $S(V) = T(V) / I$ where $I = \generators{x\tensor y - y\tensor x} \normal T(\lieg)$.

*Some facts:*

a. There is a natural grading $S(V) = \bigoplus_{n\in\NN} S^n(V)$ where $S^0(V) = F, S^1(V) = V, S^n(V) = T^n(V) / (I \intersect T^n V)$,
b. If $\theset{x_i}^n$ is a basis of $V$, then $S(V) \cong F[x_1, \cdots, x_n]$.

### Construction of UEA

**Definition:**
For $\lieg$ a lie algebra, define $U(\lieg) = T(\lieg)/J$ where $J = \generators{x\tensor y - y\tensor x - [x,y]} \normal T(\lieg)$.

Thus we have the following type of equation that holds in $U(\lieg)$:

\begin{align*}
v_1 \tensor \cdots \tensor v_a \tensor (x\tensor y) \tensor w_1 \tensor \cdots \tensor w_b = \\
v_1 \tensor \cdots \tensor v_a \tensor (y\tensor x) \tensor w_1 \tensor \cdots \tensor w_b +
v_1 \tensor \cdots \tensor v_a \tensor ([x,y]) \tensor w_1 \tensor \cdots \tensor w_b
.\end{align*}

**Proposition:**
$U(\lieg)$ has a universal property: given a lie algebra hom $\theta: \lieg \to \mathcal A$ where $\mathcal A$ is any unital associative $F\dash$algebra with a lie bracket,
there exists a unique $\psi:U(\lieg) \to \mathcal A$ making the following diagram commute:

\begin{center}
\begin{tikzcd}
\lieg \arrow[rrdd, "\theta"] \arrow[rr, "\iota"] &  & U(\lieg) \arrow[dd, "\exists \psi", dotted] \\
                                                 &  &                                             \\
                                                 &  & \mathcal A                                 
\end{tikzcd}
\end{center}


where $\iota: \lieg \injects T(\lieg) \surjects U(\lieg)$ is given by  $x\mapsto x + J$.

The upshot:
There is a 1 to 1 correspondence 

\begin{align*}
\left\{ \parbox[c]{1.1in}{\centering
  Lie algebra representations $\lieg \to \liegl(V)$
}\right\}
&\rightarrow
\left\{ \parbox[c]{1.1in}{\centering
  Algebras from $U(\lieg) \to \mathrm{End}(V)$
}\right\} \\
\theta &\mapsto \psi \\
\theta = \psi\circ\iota &\mapsfrom \psi
\end{align*}

*Proof (existence):*

$\theta: \lieg \to \mathcal A$ extends to an algebra homomorphism $\tilde\theta: T(\lieg) \to \mathcal A$ given by $\tensor_{i=1}^n x_i \mapsto \prod \theta(x_i)$.
Note that $\tilde\theta(x\tensor y - y\tensor x - [x,y]) = \theta(x)\theta(y) - \theta(y)\theta(x) - \theta([x,y]) = 0$,
and thus $J \normal \ker \tilde\theta$ and $\phi: T(\lieg)/J \to \mathcal A$ is well-defined.

*Proof (uniqueness):*
Suppose that $\psi': U(\lieg) \to \mathcal A$ is another hom $\psi'$ such that the following diagram commutes:

\begin{center}
\begin{tikzcd}
\lieg \arrow[rrdd, "\theta"] \arrow[rr, "\iota"] &  & U(\lieg) \arrow[dd, "\psi", bend left] \arrow[dd, "\psi'", bend right] \\
                                                 &  &                                                                        \\
                                                 &  & \mathcal A                                                            
\end{tikzcd}
\end{center}

Since $T(\lieg)$ is generated by $T^1(\lieg)$, $U(\lieg)$ is generated by $\iota(\lieg) \in U(\lieg)$.
Thus for all $x\in \lieg$, $\psi\circ \iota(x) = \theta(x) = \psi' \circ \iota(x)$ by the commuting of each triangle.
We then have $\psi = \psi'$ on $\iota(\lieg)$, and hence on $U(\lieg)$.


### PBW Theorem

> PBW: Poincaré-Birkhoff-Witt

**Theorem:**
If $\lieg$ has a basis $\theset{x_i}_{i\in I}$ where $\leq$ is a total order on $I$, then let $y_i \coloneqq \iota(x_i) \in U(\lieg)$.
Then $U(\lieg)$ has an $F\dash$basis called a *PBW basis* which is given by 

\begin{align*}
\theset{ y_{i_1}^{r_1} \cdots y_{i_n}^{r_n} \mid n\in \NN, r_i \in \NN, i_1 \leq \cdots \leq i_n}
.\end{align*}

We refer to each term appearing as a *PBW monomial*.

*Examples:*

Type A, $\lieg = \liesl(2, F) = \generators{f, h, e}$.
Pick an order $x_1 = f, x_2 = h, x_3 = e$, so $f < h < e$.

Then $U(\lieg)$ has a basis

\begin{align*}
B = \theset{1} \union 
\theset{f^{r^1}} \theset{f^{r^1} h^{r_2}} \union \theset{f^{r_1} h^{r_2} e^{r_3}} \union
\theset{h^{r_1}} \union \theset{f^{r_1}e^{r_2}} \union
\theset{e^{r_1}} \union \theset{h^{r_1} e^{r_2}}
.\end{align*}

i.e. $B = \theset{f^a h^b e^c \suchthat a,b,c \in \NN}$.

If you pick a different order, say $f < e < h$, then we obtain
$B = \theset{f^a e^b h^c \suchthat a,b,c \in \NN}$.