# Starting Kumar (Friday, August 27)

## 1.1: Definition of Kac-Moody Algebras

:::{.definition title="Realization"}
Let $A\in \Mat(\ell\times\ell, \CC)$ be rank $r$.
A **realization** of $A$ is a triple $(\lieh, \pi, \pi\dual)$ where $h\in\mods{\CC}$, $\pi = \ts{\alpha_1, \cdots, \alpha_\ell} \subseteq \lieh\dual$ are column vectors, and \( \ts{ \alpha_1\dual, \cdots, \alpha_\ell\dual } \subseteq \lieh \) are row vectors are indexed sets satisfying

1.  $\pi, \pi\dual$ are linearly independent sets.
2. $\alpha_j( \alpha_i\dual) = a_{i, j}$
3. $\ell - r = \dim_\CC(\lieh) - \ell$

:::

:::{.proposition title="?"}
There exists a realization of $A$ that is unique up to isomorphism.
Moreover, realizations of $A, B$ are isomorphic iff $B$ is similar to $A$ via a permutation of the index set.
:::

:::{.proof title="?"}
Assume $A$ is of the form 
\[
A = 
\begin{bmatrix}
A_1  
\\
A_2
\end{bmatrix}
,\]
where $A_1$ is $r\times \ell$ block where $\rank A_1 = r$ and $A_2$ is $l-r\times \ell$
Set 
\[
C \da 
\begin{bmatrix}
A_1 & 0 
\\
A_2 & I_{\ell-r}
\end{bmatrix}\in \Mat(\ell \times (2\ell - r))
.\]
For $\lieh = \CC^{2\ell - r}$, set $\alpha_1, \cdots, \alpha_\ell$ to be the first $\ell$ coordinate functions $\alpha_1\dual,\cdots$ as the rows of $C$.
This is a realization.

Conversely, given a realization $(\lieh, \pi, \pi\dual)$, we can produce a matrix: complete $\pi$ to a basis of $\lieh\dual$.
This can done in such a way that $\alpha_j(\alpha_i\dual) = [A_1, B; A_2, D\inv]\in \Mat(\ell \times 2\ell -r$.
Using column operations, i.e. multiplication on the right, this can be mapped to $[A_1, 0; A_2, I]$.
:::

:::{.definition title="Free Lie algebra generated by a vector space"}
Let $V\in \mods{\CC}$ and $T^\bullet(V)$ be its (associative) tensor algebra.
Set $[ab] = ab-ba$ and take $F(V) \subseteq T(V)$ to be the free Lie algebra generated by $T^1(V)$.
We call $F(V)$ the **free Lie algebra generated by $V$**.
There is a universal property: for any linear hom $\theta: V\to \liesl$, there is a commuting diagram

\begin{tikzcd}
	V && {\mathfrak{s}} \\
	\\
	& {F(V)}
	\arrow["\theta", from=1-1, to=1-3]
	\arrow["F"', from=1-1, to=3-2]
	\arrow["{\exists\tilde \theta}"', dashed, from=3-2, to=1-3]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsMyxbMCwwLCJWIl0sWzIsMCwiXFxtYXRoZnJha3tzfSJdLFsxLDIsIkYoVikiXSxbMCwxLCJcXHRoZXRhIl0sWzAsMiwiRiIsMl0sWzIsMSwiXFxleGlzdHNcXHRpbGRlIFxcdGhldGEiLDIseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XV0=)

Note that $U(F(V)) = T(V)$.

This can be constructed as 
\[
\lieh \oplus \gens{e_1, \cdots, e_\ell} \bigoplus \gens{f_1, \cdots, f_\ell} / \sim
\\ \\
\sim \da
\begin{cases}
[e_i f_i] = \delta_{ij} \alpha_i\dual & i, j = 1,\cdots, \ell 
\\
[hh'] = 0 & h,h'\in\lieh
\\
[he_i = \alpha_i(h)e_i & 
\\
[hf_i = \alpha_i(h)f_i & i=1,\cdots,\ell, h\in \lieh
\end{cases}
.\]
Then set $\tilde \lieg(A) \da F(V)/\sim$
We'll find that this only depends on the realization of $A$.
:::

:::{.definition title="Generalized Cartan Matrices"}
A matrix $A = (\alpha_{ij})$ is a **generalized Cartan matrix (GCM)**:

- $\alpha_{ii} = 2$
- $\alpha_{ij} \leq 0, i\neq j$
- $\alpha_{ij} = 0$ if $\alpha_{ji} =0$

:::

:::{.definition title="Kac-Moody Lie Algebras"}
The **Kac-Moody Lie algebra** is defined by $\lieg \da \lieg(A) \da \tilde\lieg(A)/\sim$, where we mod out by the **Serre relations**:
\[
(\ad e_i)^{1 - a_{ij}}(e_j) &= 0\\
(\ad f_i)^{1 - a_{ij}}(f_j) &= 0
.\]
:::

:::{.remark}
\envlist

- There is an injection $\lieh \injects \lieg$, so we refer to $\lieh$ as the **Cartan subalgebra**.
- The $e_i, f_i$ are **Chevalley generators**.
- The **nilradicals** are $\lien \da \gens{\ts{e_1,\cdots,e_\ell}}$ and $\lien^- \da \gens{ \ts{f_1, \cdots, f_\ell}}$.
- $\lieb \da \lieh \oplus \lien$ is the **standard Borel**.
- $\lieb^- \da \lieh \oplus \lien$
- $\tilde \lien, \tilde \lien^-, \tilde \lieb, \tilde \lieb^-$ can similarly be defined for $\tilde \lieg$.

:::

:::{.remark}
A big theorem from algebraic groups: a connected reductive group $G$ corresponds to a root datum$(\lieg, \ts{\alpha_i}_{i\leq \ell}, \ts{ \alpha_i\dual }_{i\leq \ell} )$ where $\alpha_i, \alpha_i\dual \in \ZZ^n$ such that $a_{ij} \da \inner{\alpha_i}{\alpha_i\dual}$ form a Cartan matrix $A \da (a_{ij})$.
:::

:::{.example title="?"}
Consider pairs of $K, G$ where $G$ is the complexification of $K$:

- $\Sp_n \leadsto \Sp_{2n}(\CC)$, $Z(G) = \ZZ/2$ for $n\geq 1$
- $\SU_n \leadsto \SL_n(\CC)$, $Z(G) = \ZZ/4n$ for $n\geq 3$
- $\Spin_n \leadsto \Spin_n(\CC)$, $Z(G) = (\ZZ/2)^2$, $n\geq 8$ even
- $F_4$, $Z(G) = \ZZ/4$ for $n\geq 7$ odd
- $G_2$
- $E_6$
- $E_7$
- $E_8$

Here we take the simply connected groups for the last 5, and the last 4 have cyclic centers.
:::

:::{.theorem title="?"}
There exist

1. Simple, simply connected, connected groups $G_1, \cdots, G_k$,
2. A finite central subgroup $F \subseteq \prod G_i \times T'$ where $T'$ is a (not necessarily maximal) torus,

such that $G\cong (\prod G_i \times T')/F$.
All connected reductive groups arise this way!
:::

:::{.example title="?"}
Let $G\da\GL_n = \SL_n \cdot \CC\units$, and they intersect at roots of unity, so 
\[
\GL_n = (\SL_n \times \CC\units) / \gens{\zeta_n I_n, \zeta_n\inv}
.\]
The map (in the reverse direction) is $(g, z)\mapsto gz$, and if $gz= I$ in $\GL_n$ then $g = \zeta_n^k I_n$ and $z = \zeta_k\inv$.
:::

:::{.remark}
Assume $G$ is semisimple, simply connected, and connected.
Then

1. The equivariant cohomology is
\[
H^*_T(G/B; \QQ) \cong S_\QQ \tensor_{S_\QQ^W} S_\QQ
\]
2. The equivariant \(\K\dash\)theory 
\[
K^T(G/B) = A(T) \tensor_{A(T)^W}A(T)
\]

Note that 
\[
W &= N_G(T)/T \\
S &= S(\lieh\dual), && \pi \subseteq \lieh\dual \\
A(T) &= \ZZ[X(T)]
.\]
:::

:::{.remark}
Think about semisimple, simply connected, and connected groups most of the semester.
:::