# Wednesday December 04

## Summary of Lie Algebras

- Overview
    - Definition of Lie Algebra, abelian, nilpotent, solvable, (semi)simple, reductive = semisimple and abelian.
  - Killing form $\kappa(x, y) = \tr(\ad x \circ \ad y)$
    - Solvable iff $\kappa(x, y) = 0$ for all $x\in [\lieg, \lieg], y\in \lieg$.
    - Semisimple iff $\kappa$ is non-degenerate
  - Interested in Kac-Moody algebras
    - Finite = semisimple Lie algebra, finite dimensional
    - Affine = infinite dimensional
    - Indefinite = hard!
- Structure theory for semisimple Lie Algebras
  - Semisimple = direct sum of simples
  - Semisimples are in 1-to-1 correspondence with Dynkin diagrams for $A_\ell \to D_\ell$ (classical) or $E_{6-8}, F_4 ,G_2$ (exceptional), which are also in 1-to-1 correspondence with Cartan matrices $A$
  - Presentations of $\lieg(A) = \generators{e_i, f_i, h_i}$ mod Cartan relations and Serre relations using $a_{ij}$
  - $\liesl(2) = \generators{e,f,h}/\sim$ where $[h, e] = 2e, [e,f] = h, [h, f] = -2f$
  - Cartan subalgebras $\lieh \definedas$ nilpotent + self-normalizing $\iff$ maximal toral subalgebra
    - By the conjugacy theorem, $\rank \lieg \definedas \dim \lieh$ is well-defined
    - By the abstract Jordan decomposition yields a root-space decomposition $\lieg = \lieh + \sum_{\alpha \in \Phi} \lieg_\alpha$
  - If $\Pi$ is a fixed set of simple roots, then there exists a triangular decomposition $\lieg = n^- \oplus \lieh \oplus n$ where $n^- = f's, \lieh = h's, n = e's$
  - Semisimple $\iff \kappa$ non-degenerate $\iff \lieh^* \cong \lieh$ by the map $\alpha \mapsto t\alpha$, where $\alpha = \kappa(t_\alpha, \wait)$
  - $(\alpha, \beta) \definedas \kappa(t_\alpha, t_\beta)$, coroots $\beta\dual = 2\beta/(\beta, \beta)$
  - $(\alpha, \beta\dual) = \kappa(t_\alpha, h_\beta) = \alpha(h_\beta)$ yields an inner product
  - Generates reflections $s_\alpha: \lieh^* \to \lieh*$ where $\lambda \mapsto \lambda - (\lambda, \alpha\dual)\alpha$
  - Yields the Weyl group $W = \generators{s_\alpha \mid \alpha \in \Pi}$
    - Every $w\in W$ has a reduced expression $w = \prod_i S_{\alpha_i}$
    - $\ell(w) =\text{ length of $w$ } = \#\theset{\alpha \in \Phi^+ \mid w\alpha \in \Phi^-}$
  - Universal enveloping algebra has a PBW basis
  - $Z(U(\lieg)) \cong \mathcal{S}(\lieh)^W$
  - Yields central characters $x_\lambda = x_\mu \iff \lambda \in W \cdot \mu$ where $w\cdot \mu = ?$

- $Z(U(\lieg)) \ni \Omega = \sum x_i x_i'$ where $\kappa(x_i, x_j') = \delta_{ij}$ the Casimir element
  - This acts on simple modules by a scalar, where $\Omega \actson M(\lambda)$ by $(\lambda+p, \lambda+p) - (p, p) = (\lambda + 2p, \lambda)$

## Representation Theory of Semisimple Lie Algebras

- Simple = irreducible modules, but simple $\neq$ indecomposable modules
- Composition series, completely reducible = direct sum of irreducibles
- Construct new modules by $V\tensor W, V\dual, \hom(V, W) = V\dual \tensor W$
- Theorem (Weyl): If $\lieg$ is semisimple, then any finite-dimensional module is completely reducible
- Integral weights $\Lambda = \sum_i \ZZ w_i$, where $w_i$ is a fundamental weight such that $(w_i, \alpha_j\dual) = \delta_{ij}$
- The dominant integral weights are given by $\Lambda^+ = \sum_i \ZZ_{\geq 0} w_i$
  - For $\lieg = \liesl(2)$, we have
    - $\lieg^* \cong \CC$
    - $\lambda \mapsto \ZZ$
    - $\alpha_1 \mapsto 2$
    - $\rho = w_j \mapsto 1$
    - Verma $M(\lambda) = \spanof(v_0, v_1, \cdots)$ corresponding to weights $\lambda, \lambda-2, \cdots -\lambda$.
    - Irreducible $L(\lambda) = \spanof(\overline v_0, \overline v_1, \cdots)$
    - Formal characters $\ch M(\lambda) = e(\lambda) + e(\lambda - 2) + \cdots \sim e(\lambda)(1 + e(-2) + e(-2)^2 + \cdots) \sim \frac{e(\lambda)}{1 - e(-2)}$ as a formal power series
    - Similarly, $\ch L(\lambda) = e(\lambda) + e(\lambda - 2) + \cdots$
- If $\lieg$ is semisimple, then there is a weight module, highest weight module, and maximal vectors
- Verma modules $M(\lambda) = \Ind_\lieh^\lieg \CC_\lambda = U(\lieg) \tensor_{U( \lieh )} \CC_\lambda$
- Yields $\mathfrak{b}$ the Borel subalgebra given by $\mathfrak{b} = n \oplus \lieh$, has basis $\theset{\vector{f}^{\vector{b}} v^+}$
- Irreducible modules $L(\lambda) = M(\lambda)/ N(\lambda)$ where $N(\lambda)$ is the sum of proper submodules of $M(\lambda)$.
- $\ch M(\lambda) = e(\lambda) / \prod_{\alpha \in \Phi^+} (1 - e(-\alpha))$
- Theorem (Weyl): If $\lambda \in \Lambda^+$, then there is a formula for $\ch L(\lambda)$.
- If $\lambda \not\in \Lambda^+$, then $\ch L(\lambda)$ can be deduced using composition multiplicity $[M(\lambda) : L(\mu)]$.
  - These are obtained from the Kazhdan-Lusztig polynomials
  - Extended to category $\mathcal{O}$


# Some Possible Generalizations

- Swap $\CC$ with $\RR$ of $\overline \FF_p$
- Finite leads to  affine or indefinite
- Lie Algebras lead to Algebraic groups/ Lie groups
- Can also consider Lie super-algebras
- Quantisation leads to quantum groups