# Monday August 12 > The material for this class will roughly come from Humphrey, Chapters 1 to 5. There is also a useful appendix which has been uploaded to the ELC system online. ## Overview Here is a short overview of the topics we expect to cover: ### Chapter 2 - Ideals, solvability, and nilpotency - Semisimple Lie algebras - These have a particularly nice structure and representation theory - Determining if a Lie algebra is semisimple using Killing forms - Weyl's theorem for complete reducibility for finite dimensional representations - Root space decompositions ### Chapter 3-4 We will describe the following series of correspondences: ```{=latex} \begin{tikzcd} \text{Semisimple algebras} \arrow[rr, Leftrightarrow] & & \text{Root systems} \arrow[rr, Leftrightarrow] & & \text{Dynkin diagrams} \\ & & & & \\ \text{Simple algebras over } \CC \arrow[uu, Rightarrow, "\bigoplus"] \arrow[rr, Leftrightarrow] & & \text{Irreducible root systems} \arrow[uu, Rightarrow, "\coprod"] \arrow[rr, Leftrightarrow] & & \arrow[uu, Rightarrow, "\coprod"] \text{Connected Dynkin diagrams} \end{tikzcd} ``` ## Classification The classical Lie algebras can be essentially classified by certain classes of diagrams: \begin{tbl}{The Dynkin diagrams of the simple root systems} \dyn{A}{} \dyn{B}{} \dyn{C}{} \dyn{D}{} \dyn{E}{6} \dyn{E}{7} \dyn{E}{8} \dyn{F}{4} \dyn{G}{2} \end{tbl} ## Chapters 4-5 These cover the following topics: - Conjugacy classes of Cartan subalgebras - The PBW theorem for the universal enveloping algebra - Serre relations ### Chapter 6 Some import topics include: - Weight space decompositions - Finite dimensional modules - Character and the Harish-Chandra theorem - The Weyl character formula - This will be computed for the specific Lie algebras seen earlier We will also see the type $A_{\ell}$ algebra used for the first time; however, it differs from the other types in several important/significant ways. ### Chapter 7 Skip! ### Topics Time permitting, we may also cover the following extra topics: - Infinite dimensional Lie algebras [Carter 05] - BGG Cat$\dash\mathcal O$ [Humphrey 08] ## Content Fix $F$ a field of characteristic zero -- note that prime characteristic is closer to a research topic. \wrapenv{\Begin{definition}} A **Lie Algebra** $\lieg$ over $F$ is an $F\dash$vector space with an operation denoted the Lie bracket, \begin{align*} [\wait, \wait]: \lieg \cross \lieg \to \lieg \\ (x,y) \mapsto [x, y] .\end{align*} satisfying the following properties: - $[\wait, \wait]$ is bilinear - $[x, x] = 0$ - The Jacobi identity: \begin{align*} [x, [y, z]] + [y, [x,z]] + [z, [x, y]] = \vector 0 .\end{align*} \wrapenv{\End{definition}} \wrapenv{\Begin{exercise}} Show that $[x, y] = -[y,x]$. \wrapenv{\End{exercise}} \wrapenv{\Begin{definition}} Two Lie algebras $\lieg, \lieg'$ are said to be isomorphic if $\varphi([x, y]) = [\varphi(x), \varphi(y)]$. \wrapenv{\End{definition}} ## Linear Lie Algebras Let $V = \FF^{n}$, and define $\mathrm{End}(V) = \theset{f: V \to V \suchthat V \text{ is linear}}$. We can then define $\liegl(n, V)$ by setting $[x, y] = (x\circ y) - (y\circ x)$. \wrapenv{\Begin{exercise}} Verify that $V$ is a Lie algebra. \wrapenv{\End{exercise}} \wrapenv{\Begin{definition}} Define $$\liesl(n, V) = \theset{f \in \liegl(n, V) \suchthat \Tr(f) = 0}.$$ (Note the different in definition compared to the lie *group* $\SL(n, V)$.). \wrapenv{\End{definition}} \wrapenv{\Begin{definition}} A *subalgebra* of a Lie algebra is a vector subspace that is closed under the bracket. \wrapenv{\End{definition}} \wrapenv{\Begin{definition}} The symplectic algebra \begin{align*} \liesp(2\ell, F) = \theset{A \in \liegl(2\ell, F)\suchthat MA-A^{T}M = 0} \text{ where } M = \left(\begin{array}{c|c}{0} & {I_{n}} \\ \hline {-I_{n}} & {0}\end{array}\right) .\end{align*} \wrapenv{\End{definition}} \wrapenv{\Begin{definition}} The orthogonal algebra \begin{align*} \lieso(2\ell, F) = \theset{A \in \liegl(2\ell, F)\suchthat MA-A^{T}M = 0} \text{ where } \\ M = \begin{cases} \left(\begin{array}{l|l} {1} & {0} \\ \hline {0} & {\begin{array}{c|c}{0} & {I_{n}} \\\hline {-I_{n}} & {0}\end{array}} \end{array}\right) & n=2\ell + 1 \text{ odd},\\ \\ \left(\begin{array}{c|c}{0} & {I_{n}} \\ \hline {-I_{n}} & {0}\end{array}\right) & \text{ else}. \end{cases} \end{align*} \wrapenv{\End{definition}} \wrapenv{\Begin{proposition}} The dimensions of these algebras can be computed; - The dimension of $\liegl(n, \FF)$ is $n^{2}$, and has basis ${\theset{e_{{i,j}}}}$ the matrices if a 1 in the $i,j$ position and zero elsewhere. ![$x$ is determined to force the trace to be zero](figures/2019-08-17-01:40.png)\ - For type $A_{\ell}$, we have $\dim \liesl(n, \FF) = (\ell+1)^{2} - 1$. - For type $C_{\ell}$, we have $\abs{}{\liesp(n, \FF)} = \ell^{2} + 2\left(\frac{\ell(\ell+1)}{2} \right )$, and so elements here \begin{align*} \left(\begin{array}{ll} {A} & {B=B^{t}} \\ {C = C^{t}} & {A^{t}} \end{array}\right) .\end{align*} \wrapenv{\End{proposition}} - For type $D_{\ell}$ we have \begin{align*} \abs{}{\lieso(2\ell, \FF)} = \dim\theset{ \left(\begin{array}{ll}{A} & {B=-B^{t}} \\ {C = -C^{t}} & {-A^{t}} \end{array}\right)} ,\end{align*} which turns out to be $2\ell^{2}-\ell$. - For type $B_{\ell}$, we have $\dim{\lieso}(2\ell, \FF) = 2\ell^{2} -\ell+2\ell = 2\ell^{2} + \ell$, with elements of the form \begin{align*} \left(\begin{array}{c|cc} 0 & M & N \\ \hline -N^{t} & A & C=C^{t} \\ -M^{t} & B=B^{t} & -A^{t} \end{array}\right) .\end{align*} \wrapenv{\Begin{exercise}} Use the relation $MA = A^{tM}$ to reduce restrictions on the blocks. \wrapenv{\End{exercise}} ```{=latex} \begin{tikzcd} && & \lieso(6) & & \\ && & \lieso(5) \arrow[rrd] & & \\ \liesl(4) \arrow[rrruu] & \liesl(2)^2 \arrow[rr] & & \lieso(4) & & \liesp(4) \\ && & \lieso(3) & & \\ & \liesl(2) \arrow[rru] & & & & \liesp(2) \arrow[llu] \end{tikzcd} ``` \wrapenv{\Begin{theorem}} These are *all* of the isomorphisms between any of these types of algebras, in any dimension. \wrapenv{\End{theorem}}