# Wednesday January 8 > Course Website: ## Chapter Zero: Review > Material can be found in Humphreys 1972. :::{.exercise title="Assignment Zero"} Practice writing lowercase mathfrak characters! ::: In this course, we'll take $k = \CC$. :::{.definition title="Lie Algebra"} Recall that a Lie Algebra is a vector space $\lieg$ with a bracket \[ [\wait, \wait]: \lieg\tensor \lieg \to \lieg \] satisfying - $[x x] = 0$ for all $x\in \lieg$ - $[x [y z]] = [[x y] z] + [y [x z]]$ (The Jacobi identity) ::: Note that the last axiom implies that $x$ acts as a derivation. :::{.exercise title="?"} Show that $[x y] = -[y x]$. > Hint: Consider $[x+y, x+y]$. > Note that the converse holds iff $\ch k \neq 2$. ::: :::{.exercise title="?"} Show that Lie Algebras never have an identity. ::: :::{.definition title="Abelian Lie Algebras"} $\lieg$ is *abelian* iff $[x y] = 0$ for all $x,y\in\lieg$. ::: There are also the usual notions (define for rings/algebras) of: - Subalgebras, - A vector subspace that is closed under brackets. - Homomorphisms - I.e. a linear transformation $\phi$ that commutes with the bracket, i.e. $\phi([x y]) = [\phi(x) \phi(y)]$. - Ideals :::{.exercise title="?"} Given a vector space (possibly infinite-dimensional) over $k$, then (exercise) $\liegl(V) \definedas \mathrm{End}_k(V)$ is a Lie algebra when equipped with $[f g] = f\circ g - g\circ f$. ::: :::{.definition title="Representation"} A *representation* of $\lieg$ is a homomorphism $\phi: \lieg \to \gl(V)$ for some $V$. ::: :::{.example title="The adjoint representation"} The adjoint representation is \[ \ad: \lieg &\to \liegl(\lieg) \\ \ad(x)(y) &\definedas [x y] .\] ::: Representations give $\lieg$ the structure of a module over $V$, where $x\cdot v \definedas \phi(x)(v)$. All of the usual module axioms hold, where now \[ [x y] \cdot v \da x\cdot(\cdot v) - y\cdot(x\cdot v) \] :::{.example title="?"} The trivial representation $V = k$ where $x\cdot a = 0$. ::: :::{.definition title="Irreducible"} $V$ is *irreducible* (or *simple*) iff $V$ as exactly two $\lieg\dash$invariant subspaces, namely $0, V$. ::: :::{.definition title="Completely Reducible Modules"} $V$ is *completely reducible* iff $V$ is a direct sum of simple modules, and *indecomposable* iff $V$ can not be written as $V = M \oplus N$, a direct sum of proper submodules. ::: There are several constructions for creating new modules from old ones: - The *contragradient/dual*: :::{.definition title="Contragradient dual"} \[ V\dual &\definedas \hom_k(V, k) \qquad (x\cdot f) &= -f(x\cdot v) .\] for $f\in V\dual, x\in \lieg, v\in V$. ::: - The direct sum $V\oplus W$ where \[ x\cdot(v, w) = (x\cdot v, x\cdot w) \] - The tensor product where \[ x\cdot(v\tensor w) = x\cdot v \tensor w + v\tensor x\cdot w \] - $\hom_k(V, W)$ where \[ (x\cdot f)(v) = x\cdot f(v) - f(x\cdot v) \] - Note that if we take $W=k$ then the first term vanishes and this recovers the dual. ## Semisimple Lie Algebras :::{.definition title="Derived Ideal"} The *derived ideal* is given by $\lieg^{(1)} \definedas [\lieg \lieg] \definedas \spanof_k\qty{\theset{[x y] \suchthat x,y\in\lieg }}$. ::: This is the analog of the commutator subgroup. :::{.lemma title="The derived ideal detects abelian algebras"} $\lieg$ is abelian iff $\lieg^{(1)} = \theset{0}$, and 1-dimensional algebras are always abelian. ::: :::{.proof title="?"} This follows because if $[x y] \definedas xy = yx$ then $[x y] = 0 \iff xy = yx$. ::: :::{.definition title="Simple algebras"} A lie algebra $\lieg$ is *simple* iff the only ideals of $\lieg$ are $0, \lieg$ and $\lieg^{(1)} \neq \theset{0}$. ::: Note that thus rules out the zero modules, abelian lie algebras, and particularly 1-dimensional lie algebras. :::{.definition title="Derived Series and Solvability"} The *derived series* is defined by $\lieg^{(2)} = [\lieg^{(1)} \lieg^{(1)}]$, continuing inductively. $\lieg$ is said to be **solvable** if $\lieg^{(n)} = 0$ for some $n$. ::: :::{.lemma title="?"} Abelian implies solvable. ::: :::{.definition title="Nilpotent Algebras"} The **lower central series** of $\lieg$ is defined as $\lieg_{j+1} \da [\lieg, \lieg_j]$. The lie algebra $\lieg$ is **nilpotent** if this series terminates at zero. ::: :::{.remark} Note that an *element* $x$ of a Lie algebra is nilpotent iff $\ad x$ is nilpotent as a matrix (so $x$ is *ad-nilpotent*), i.e. $\ad(x)^n =0$ for some $n$. There is a result, Engel's theorem, which relates these two notions: a Lie algebra is nilpotent iff all of its elements are nilpotent (with potentially different $n$s depending on $x$). ::: :::{.definition title="Semisimple"} $\lieg$ is *semisimple* (s.s.) iff $\lieg$ has no nonzero solvable ideals. ::: :::{.exercise title="?"} Show that simple implies semisimple. ::: :::{.remark} \envlist 1. Semisimple algebras $\lieg$ will usually have solvable subalgebras. 2. $\lieg$ is semisimple iff $\lieg$ has no nonzero abelian ideals. ::: :::{.definition title="Killing Form"} The *Killing form* is given by $\kappa: \lieg \tensor \lieg \to k$ where $\kappa(x, y) = \tr(\ad x ~\ad y)$, which is a symmetric bilinear form. ::: :::{.lemma title="?"} \[ \kappa([x y], z) = \kappa(x, [y z]) \] ::: :::{.definition title="Radical"} If $\beta: V^{\tensor 2} \to k$ is any symmetric bilinear form, then its radical is defined by \[ \rad \beta = \theset{v\in V \suchthat \beta(v, w) = 0 ~\forall w\in V} .\] ::: :::{.definition title="Nondegenerate Bilinear Forms"} A bilinear form $\beta$ is *nondegenerate* iff $\mathrm{rad}\beta = 0$. ::: :::{.lemma title="?"} $\mathrm{rad}\kappa \normal \lieg$ is an ideal, which follows by the above associative property. ::: :::{.theorem title="Characterization of Semisimplicity Using the Killing Form"} $\lieg$ is semisimple iff $\kappa$ is nondegenerate. ::: :::{.example title="?"} The standard example of a semisimple lie algebra is \[ \lieg = \liesl(n, \CC) \definedas \theset{x\in \liegl(n, \CC) \suchthat \tr(x) = 0 } \] ::: :::{.remark} From now on, $\lieg$ will denote a semisimple lie algebra over $\CC$. ::: :::{.theorem title="Weyl's Complete Reducibility Criterion"} Every finite dimensional representation of a semisimple $\lieg$ is completely reducible. ::: In other words, the category of finite-dimensional representations is relatively uninteresting -- there are no extensions, so everything is a direct sum. Thus once you classify the simple algebras (which isn't terribly difficult), you have complete information.