# Monday January 13th ## Lengths Recall that we have a root space decomposition $\lieg = \lieh \oplus \bigoplus_{\beta \in \Phi} \lieg_\beta$ for finite dimensional semisimple lie algebras over $\CC$. We have $s_\beta(\lambda) = \lambda - (\lambda, \beta\dual)\beta$, for $\lambda \in \lieh\dual$ and the Weyl group $$ W = \generators{s_\beta \suchthat \beta\in\Phi} = \generators{s_\alpha \suchthat \alpha \in \Delta} $$ where $\Delta = \theset{a_i}$ are the simple roots. For $w\in W$, we can take the reduced expression for $w$ by writing $w = s_1 \cdots s_n$ with $s_i$ simple and $n$ minimal. The length is uniquely determined, but not the expression. So we define $\ell(w) \definedas n$ where $\ell(1) \definedas 0$. :::{.fact} \envlist 1. $\ell(w)$ is the size of the set $\theset{\beta\in\Phi^+ \suchthat w\beta < 0}$ - The above set is equal to $\Phi^+ \intersect w\inv \Phi^-$. - In particular, for $\beta \in \Phi^+$, $\beta$ is simple (i.e. $\beta \ni \Delta$ iff $\ell(s_\beta) = 1)$. - Note: $\alpha$ is the only root that $s_\alpha$ sends to a negative root, so $s_\alpha(\beta) > 0$ for all $\beta\in\Phi^+\setminus\theset{\alpha}$. 2. $\ell(w) = \ell(w\inv)$ for all $w\in W$, so $\ell(w)$ is also the size of $\Phi \intersect w\Phi$ (replacing $w\inv$ with $w$) 3. There exists a unique $w_0 \in W$ with $\ell(w_0)$ maximal such that $\ell(w_0) = \abs{\Phi^+}$ and $w_0(\Phi^+) = \Phi^-$. - Also $\ell(w_0 w) = \ell(w_0) - \ell(w)$ [^1] 4. For $\alpha \in \Phi^+$, $w\in W$, we have either \[ \ell(w s_\alpha) > \ell (w) \iff w(\alpha) > 0 \\ \ell(w s_\alpha) < \ell (w) \iff w(\alpha) < 0 \\ .\] Taking inverses yields $\ell(s_\alpha w) > \ell(w) \iff w\inv\alpha > 0$. [^1]: Note that the product of reduced expressions is not usually reduced, so the length isn't additive. ::: ## Bruhat Order :::{.definition title="Bruhat Order"} Let $S$ be the set of simple reflections, i.e. $S = \theset{s_\alpha \suchthat \alpha \in \Delta}$. Then define $$ T \definedas \Union_{w\in W} wSw\inv = \theset{s_\beta \suchthat \beta\in\Phi^+} .$$ This is the set of *all* reflections in $W$ through hyperplanes in $E$. We'll write $w' \mapsvia{t} w$ means $w=tw'$ and $\ell(w') < \ell(w)$. Note that in the literature, it's also often assumed that that $\ell(w') = \ell(w) - 1$. In this case, we say $w'$ **covers** $w$, and refer to this as **the covering relation**. So $w' \to w$ means that $w' \mapsvia{t} w$ for some $t\in T$. We extend this to a partial order: $w' < w$ means that there exists a $w$ such that $$ w' = w_0 \to w_1 \to \cdots \to w_n = w. $$ This is called the **Bruhat-Chevalley order** on $W$. ::: :::{.corollary title="?"} $w' < w \implies \ell(w') < \ell(w)$, so $1\in W$ is the unique minimal element in $W$ under this order. ::: It turns out that if we set $w = w' t$ instead, this results in the same partial order. If you restrict $T$ to simple reflections, this yields the **weak Bruhat order** In this case, the left and right versions differ, yielding the **left/right weak Bruhat orders** respectively. [^2] [^2]: Note that this is because conjugating a simple reflection may not yield a simple reflection again. Recall that lie algebras yield finite crystallographic coxeter groups. :::{.proposition title="Properties of the Bruhat Order"} For $(W, S)$ a coxeter group, a. $w' \leq w$ iff $w'$ occurs as a subexpression/subword of every reduced expression $s_1 \cdots s_n$ for $w$, where a subexpression is any subcollection of $s_i$ in the same order.[^3] [^3]: Note that this implies that $1$ is not only a minimal element in this order, but an infimum. b. Adjacent elements $w', w$ (i.e. $w' < w$ and there does not exist a $w''$ such that $w' < w'' < w$) in the Bruhat order differ in length by 1. c. If $w' < w$ and $s\in S$, then $w' s \leq w$ or $w's \leq ws$ (or both). i.e., if $\ell(w_1) = 2 = \ell(w_2)$, then the size of $\theset{w\in W \suchthat w_1 < w < w_2}$ is either 0 or 2. \begin{tikzpicture} \node (0) at (0, 2) [label=$w_2$]{}; \node (1) at (2, 0) {}; \node (2) at (-2, 0) {}; \node (3) at (0, -2) [label=below:$w_1$]{}; \draw (0.center) to (2.center); \draw (2.center) to (3.center); \draw (0.center) to (1.center); \draw (1.center) to (3.center); \end{tikzpicture} ::: ## Properties of Universal Enveloping Algebras Let $\lieg$ be any lie algebra, and $\phi: \lieg \to A$ be any map into an associative algebra. Then there exists an object $U(\lieg)$ and a map $i$ such that the following diagram commutes: \begin{tikzcd} & & U(\lieg) \arrow[dd, "\tilde \phi", dotted] \\ \lieg \arrow[rrd, "\phi"] \arrow[rru, "i"] & & \\ & & A \end{tikzcd} where $\tilde \phi$ is a map in the category of associative algebras. Moreover any lie algebra homomorphism $\lieg_1 \to \lieg_1$ induces a morphism of associative algebras $U(\lieg_1) \to U(\lieg_2)$, where $\lieg$ generates $U(\lieg)$ as an algebra. $U(\lieg)$ can be constructed as $$ U(\lieg) = T(\lieg)/ \generators{ [x,y] - x\tensor y - y\tensor x \suchthat x,y\in\lieg} .$$ Note that this ideal is not necessarily homogeneous. :::{.proposition title="Properties of the Universal Enveloping Algebra"} \envlist - Usually noncommutative - Left and right Noetherian - No zero divisors - $\lieg \actson U(\lieg)$ by the extension of the adjoint action, $(\ad x)(u) = xu - ux$ for $x\in \lieg, u\in U(\lieg)$. ::: :::{.theorem title="Poincaré-Birkhoff-Witt (PBW"} If $\theset{x_1, \cdots x_n}$ is a basis for $\lieg$, then $\theset{x_1^{t_1}, \cdots, x_n^{t_n} \suchthat t_i \in \ZZ^+}$ (noting that $x^n = x\tensor x \tensor \cdots x$ and $\ZZ^+$ includes 0) is a basis for $U(\lieg)$. ::: :::{.corollary title="?"} $i:\lieg \to U(\lieg)$ is injective, so we can think of $\lieg \subseteq U(\lieg)$. If $\lieg$ is semisimple, then it admits a triangular decomposition $\lieg = \lien^- \oplus \lieh \oplus \lien$ and choosing a compatible basis for $\lieg$ yields $U(\lieg) = U(\lien^-)\tensor U(\lieh) \tensor U(\lien)$. If $\phi: \lieg \to \gl(V)$ is any Lie algebra representation, it induces an algebra representation $U(\lieg)$ of $U(\lieg)$ on $V$ and vice-versa. It satisfies $$x\cdot (y \cdot v) - y\cdot (x \cdot v) = [x y] \cdot v$$ for all $x,y \in \lieg$ and $v\in V$. ::: :::{.remark} Note that this lets us go back and forth between Lie algebra representations and associative algebra representations, i.e. the theory of modules over rings. ::: :::{.remark} A note on notation: $\mathcal Z(\lieg)$ denotes the center of $U(\lieg)$. ::: ## Integral Weights We have a Euclidean space $E = \RR \Phi^+$, the $\RR\dash$span of the roots. :::{.definition title="Integral Weight Lattice"} We also have the **integral weight lattice** $$ \Lambda = \theset{\lambda \in E \suchthat (\lambda, \alpha\dual) \in \ZZ ~\forall \alpha\in\Phi (\text{or}~ \Phi^+ ~\text{or}~ \Delta)} .$$ ::: :::{.definition title="Ordering of Weights"} There is a sublattice $\Lambda_r \subseteq \Lambda$, which is an additive subgroup of finite index. There is a partial order of $\Lambda$ on $E$ and $\lieh\dual$. We write \[ \mu \leq \lambda \iff \lambda - \mu \in \ZZ^+ \Delta = \ZZ^+ \Phi^+ \] ::: :::{.definition title="Dominant Integral Weights"} For a basis $\Delta = \theset{\alpha_1, \cdots, \alpha_n}$, define a dual basis $(w_i ,\alpha_j\dual) = \delta_{ij}$. The fundamental weights are given by a $\ZZ\dash$basis for $\Lambda$. Then $\Lambda$ is a free abelian group of rank $\ell$, and \[ \Lambda^+ = \ZZ^+ w_1 + \cdots + \ZZ^+ w_\ell \] are the **dominant integer weights**.[^5] [^5]: Note that in Jantzen's book, $X$ is used for $\Lambda$ and $X^+$ correspondingly. :::