# Useful Tricks

- $[[xy]z] = [x[yz]] + [[xz]y] = [x[yz]] - [y[xz]]$.
- $xz.w = zx.w + [xz].w$.
- If $N, M$ are upper triangular, $[NM]$ has zeros along the diagonal.

# Summary of main results

## Computation

- Bracketing elementary matrices:
\[
[e_{ij}, e_{kl}] = \delta_{jk} e_{il} - \delta_{li} e_{kj}
.\]


## Classical Algebras

- $\liesl_2(\FF)$ is dimension 3, corresponds to type $A_2$, and generated by 

\[
x=\left(\begin{array}{ll}0 & 1 \\ 0 & 0\end{array}\right)\quad
h=\left(\begin{array}{cc}
1 & 0 \\
0 & -1
\end{array}\right)
y=\left(\begin{array}{ll}0 & 0 \\ 1 & 0\end{array}\right)\quad \\ \\
[x, y]=h, \quad[h, x]=2 x, \quad[y, h]=2 y
.\]
- $\liesl_n(\FF)$ is dimension $n^2-1$ and corresponds to type $A_{n-1}$.


## Definitions

- $N_L(S) = \ts{x\in L \st [xS] \subseteq S}$.
- $C_L(S) = \ts{x\in L \st [x S] = 0}$.
- $L$ is simple iff $L\neq 0$ and $\Id(L) = \ts{0, L}$.
- $L$ is solvable iff $L^{(n+1)} \da [ L^{(n)} L^{(n)}] \converges{n\to\infty}\to 0$.
- $L$ is semisimple iff $\Rad(L) \da \sum_{I \normal L \text { solvable}} I = 0$.
- $L$ is nilpotent iff $L^{n+1} \da [L L^{n-1}] \converges{n\to\infty}\to 0$.
- The Killing form is $\kappa(x,y)\da \Trace(\ad_x \circ \ad_y)$.
- Checking $\lmod$ actions: 
  - $( \lambda_1 x + \lambda_2 y).\vector v = \lambda_1 (x.\vector v) + \lambda_2(y.\vector v)$
  - $x.(\lambda_1 \vector v+ \lambda_2 \vector w) = \lambda_1 (x.\vector v) + \lambda_2(x.\vector w)$
  - $[xy].\vector v = x.(y.\vector v) - y.(x.\vector v)$.
    This is the axiom that introduces weird formulas for homs/tensors/duals.
- Duals: $(x.f)(\vector v) \da -f(x.\vector v)$
- Tensors: $x.(\vector v\tensor \vector w) \da \qty{ (x.\vector v)\tensor \vector w} + \qty{\vector v\tensor (x.\vector w)}$.
- The Casimir element: for $\phi: L\to \liegl(V)$ an irreducible representation of $L$ semisimple, define $\beta(x, y) \da \Trace(\phi(x)\circ \phi(y))$. Pick a basis and dual basis $\ts{e_i}, \ts{e_i\dual}$ with respect to $\beta$ and define $c_\phi(\beta) \da \sum \phi(e_i)\phi(e_i\dual) \in \Endo(V)$.
  - This is an endomorphism of $V$ commuting with the $L\dash$action which has nonzero trace.


## Results

- Engel's theorem: if every $x\in L$ is ad-nilpotent then $L$ is nilpotent.
  - Since conversely (for free) $L$ nilpotent implies $\ad_x$ is nilpotent for every $x$, this becomes an iff: $L$ is nilpotent iff every $x\in L$ is ad-nilpotent.
- Lie's theorem: if $L\leq \liegl(V)$ is solvable then $L$ stabilizes a flag in $V$ (i.e. $L$ has an upper triangular basis).
- Cartan's criterion: if $L\leq \liegl(V)$ and $\Trace(xy) = 0$ for all $x\in [LL]$ and $y\in L$, then $L$ is solvable.
- $L$ is semisimple iff $\kappa_L$ is nodegenerate.
- If $L$ is semisimple, it decomposes as $L = \bigoplus L_i$ where the $L_i$ are uniquely determined simple ideals, and every simple ideal of $L$ is one such $L_i$.
- $\ker (L \mapsvia{\ad_L} \liegl(V)) = Z(L)$ and simple algebras are centerless, so any simple Lie algebra is isomorphic to a linear Lie algebra $\liegl(V)$ for some $V$, namely $\im \ad_L$.
- Schur's lemma: if $L \mapsvia{\phi} \liegl(V)$ is an irreducible representation, then $C_{\liegl(V)}(\phi(L)) = \CC I$, i.e. the only endomorphisms of $V$ commuting with every $\phi(x)$ are scalar operators.
  Equivalently, $\Endo_L(V) \cong \CC$.
- Weyl's theorem: if $L$ is semisimple and $\phi: L\to \liegl(V)$ is a finite-dimensional representation then $\phi$ is completely reducible.