$\require{physics}\newcommand{\cbrt}[1]{\sqrt[3]{#1}}\newcommand{\sgn}{\text{sgn}}\newcommand{\ii}[1]{\textit{#1}}\newcommand{\eps}{\varepsilon}\newcommand{\EE}{\mathbb E}\newcommand{\PP}{\mathbb P}\newcommand{\Var}{\mathrm{Var}}\newcommand{\Cov}{\mathrm{Cov}}\newcommand{\pperp}{\perp\kern-6pt\perp}\newcommand{\LL}{\mathcal{L}}\newcommand{\pa}{\partial}\newcommand{\AAA}{\mathscr{A}}\newcommand{\BBB}{\mathscr{B}}\newcommand{\CCC}{\mathscr{C}}\newcommand{\DDD}{\mathscr{D}}\newcommand{\EEE}{\mathscr{E}}\newcommand{\FFF}{\mathscr{F}}\newcommand{\WFF}{\widetilde{\FFF}}\newcommand{\GGG}{\mathscr{G}}\newcommand{\HHH}{\mathscr{H}}\newcommand{\PPP}{\mathscr{P}}\newcommand{\Ff}{\mathcal{F}}\newcommand{\Gg}{\mathcal{G}}\newcommand{\Hh}{\mathbb{H}}\DeclareMathOperator{\ess}{ess}\newcommand{\CC}{\mathbb C}\newcommand{\FF}{\mathbb F}\newcommand{\NN}{\mathbb N}\newcommand{\QQ}{\mathbb Q}\newcommand{\RR}{\mathbb R}\newcommand{\ZZ}{\mathbb Z}\newcommand{\KK}{\mathbb K}\newcommand{\SSS}{\mathbb S}\newcommand{\II}{\mathbb I}\newcommand{\conj}[1]{\overline{#1}}\DeclareMathOperator{\cis}{cis}\newcommand{\abs}[1]{\left\lvert #1 \right\rvert}\newcommand{\norm}[1]{\left\lVert #1 \right\rVert}\newcommand{\floor}[1]{\left\lfloor #1 \right\rfloor}\newcommand{\ceil}[1]{\left\lceil #1 \right\rceil}\DeclareMathOperator*{\range}{range}\DeclareMathOperator*{\nul}{null}\DeclareMathOperator*{\Tr}{Tr}\DeclareMathOperator*{\tr}{Tr}\newcommand{\id}{1\!\!1}\newcommand{\Id}{1\!\!1}\newcommand{\der}{\ \mathrm {d}}\newcommand{\Zc}[1]{\ZZ / #1 \ZZ}\newcommand{\Zm}[1]{\left(\ZZ / #1 \ZZ\right)^\times}\DeclareMathOperator{\Hom}{Hom}\DeclareMathOperator{\End}{End}\newcommand{\GL}{\mathbb{GL}}\newcommand{\SL}{\mathbb{SL}}\newcommand{\SO}{\mathbb{SO}}\newcommand{\OO}{\mathbb{O}}\newcommand{\SU}{\mathbb{SU}}\newcommand{\U}{\mathbb{U}}\newcommand{\Spin}{\mathrm{Spin}}\newcommand{\Cl}{\mathrm{Cl}}\newcommand{\gr}{\mathrm{gr}}\newcommand{\gl}{\mathfrak{gl}}\newcommand{\sl}{\mathfrak{sl}}\newcommand{\so}{\mathfrak{so}}\newcommand{\su}{\mathfrak{su}}\newcommand{\sp}{\mathfrak{sp}}\newcommand{\uu}{\mathfrak{u}}\newcommand{\fg}{\mathfrak{g}}\newcommand{\hh}{\mathfrak{h}}\DeclareMathOperator{\Ad}{Ad}\DeclareMathOperator{\ad}{ad}\DeclareMathOperator{\Rad}{Rad}\DeclareMathOperator{\im}{im}\renewcommand{\BB}{\mathcal{B}}\newcommand{\HH}{\mathcal{H}}\DeclareMathOperator{\Lie}{Lie}\DeclareMathOperator{\Mat}{Mat}\DeclareMathOperator{\span}{span}\DeclareMathOperator{\proj}{proj}$ This is a [[homepage]] for Lie algebras. This can be learned independently from [[Lie/Lie Groups]]. [[A story of Liers]] (WIP!) >[!idea] >Lie algebras are all taken to be finite dimensional by default! >[!idea] >Writing $\fg$ gets tiring if objects are all algebras by default. So we'll just write $L$. - [[Lie Algebra]] # Enumerating the Lie Algebras When enumerating the operators over a vector space, we diagonalize and discuss eigenvalues. When enumerating the lie algebras (which are just $(2,1)$ tensors!), we want to diagonalize somehow. The way is called a Cartan Subalgebra. - [[Cartan Subalgebra]] - [[Representation Theory Words]] - [[Casimir Element]] - [[Weyls Theorem]] - [[SL2 Worked example]] - [[Root Space Decomposition]] Everything else: - [[Classification of Simple Lie Algebras]] - See [[Simple Lie Algebra Data Page]] Quantum Field Theory Crash-Course, for fun Wait a minute, I need to know this. - [[Intuitive rephrasing with Physics]] Connecting Root systems harder to Lie algebras: - [[RS-LA Isomorphism Theorem]] # Enumerating the Representations - [[Verma Module]] - [[Characters in Lie Groups]] - [[Weyl Character Formula]] - [[Weyl Dimension Formula]] - [[All finite irreps are main diagonals]] In addition to the WCF, the main scaffold to classify representations of complex simple Lie algebras are: - [[Minuscule Weights]] (Inessential, can be read during/after $\GL_n(\CC)$) # Representation Theory of $\GL_n(\CC)$ Young Tablueax, the things you saw at the end of QFT2, they're all here. We really got to learn this. - [[Representations of SL]] - [[Representations of GL]] - [[Schur-Weyl Duality]] - [[Schur Functors]] - [[Frobenius Character Formula]] - [[Howe Duality]] - [[Fundamental Theorem of Invariant Theory]] # Representation Theory of other Simple Lie Algebras Todo: discuss $B_n, C_n, D_n$ representations. - [[Clifford Algebra Construction]] # Integration on Lie Groups See latex notes on differential geometry for a review of differential forms et al. Remember the representation theory of finite groups, with the characters and stuff? Yeah, let's run it back. - [[Haar measure for Lie Groups]] - [[Representations of Compact Lie Groups]] # All about Real Forms Up until now, we have worked solely over the field $\CC$. In order to say things about $\RR$, we follow the usual algebraic strategy. First, we'll discuss in full generality of Galois extensions of fields $L$ over $K$; then we'll specialize to real forms. # Topology of Lie Groups and Homogenous Spaces Uh oh :) Let's see if my [[Homology Theory]] self-studying is useful at all. - [[Chevalley-Eilenberg Complex]]