Talk on Chevalley's integral forms

introduction

motivating questions

why do we want integral forms?
what are good bases?
$\mathfrak{g}_{\mathbb{Z}}$ is a Lie algebra over $\mathbb{Z}$

Serre's relations

틀:수학노트 에서 가져옴
l : 리대수 $\mathfrak{g}$의 rank
$(a_{ij})$ : 카르탄 행렬
생성원 $e_i,h_i,f_i , (i=1,2,\cdots, l)$
세르 관계식
- $\left[h_i,h_j\right]=0$
- $\left[e_i,f_j\right]=\delta _{i,j}h_i$
- $\left[h_i,e_j\right]=a_{i,j}e_j$
- $\left[h_i,f_j\right]=-a_{i,j}f_j$
- $\left(\text{ad} e_i\right){}^{1-a_{i,j}}\left(e_j\right)=0$ ($i\neq j$)
- $\left(\text{ad} f_i\right){}^{1-a_{i,j}}\left(f_j\right)=0$ ($i\neq j$)
ad 는 adjoint 의 약자
- $\left(\text{ad} x\right){}^{3}\left(y\right)=[x, [x, [x, y]]]$
- $\left(\text{ad} x\right){}^{4}\left(y\right)=[x, [x, [x, [x, y]]]]$

sl(3)의 예

카르탄 행렬\[\left( \begin{array}{cc} 2 & -1 \\ -1 & 2 \end{array} \right)\]
$i\neq j$ 일 때\[\left(\text{ad} e_i\right){}^{2}\left(e_j\right)=[e_i, [e_i,e_j]]=0\]\[\left(\text{ad} f_i\right){}^{2}\left(f_j\right)=[f_i, [f_i,f_j]]=0\]
$e_1,e_2,h_1,h_2,f_1,f_2, \left[e_1,e_2\right], \left[f_1,f_2\right]$는 리대수의 기저가 된다

UEA 에서의 관계식

카르탄행렬이 $(a_{ij})$ 로 주어지는 리대수 $\mathfrak{g}$의 UEA $U(\mathfrak{g})$ 에서 다음의 두 식

\[\left(\text{ad} e_i\right){}^{1-a_{i,j}}\left(e_j\right)=0, \quad i\neq j\] \[\left(\text{ad} f_i\right){}^{1-a_{i,j}}\left(f_j\right)=0, \quad i\neq j\]

다음과 같이 표현할 수 있다\[\sum_{k=0}^{1-a_{i,j}}(-1)^k \binom{1-a_{i,j}}{k}e_{i}^{1-a_{i,j}-k}e_{j}e_{i}^k=0\]\[\sum_{k=0}^{1-a_{i,j}}(-1)^k \binom{1-a_{i,j}}{k}f_{i}^{1-a_{i,j}-k}f_{j}f_{i}^k=0\]
풀어 쓰면 다음과 같은 형태가 된다\[x\otimes x\otimes y-2 x\otimes y\otimes x+y\otimes x\otimes x\]\[x\otimes x\otimes x\otimes y-3 x\otimes x\otimes y\otimes x+3 x\otimes y\otimes x\otimes x-y\otimes x\otimes x\otimes x\]\[x\otimes x\otimes x\otimes x\otimes y-4 x\otimes x\otimes x\otimes y\otimes x+6 x\otimes x\otimes y\otimes x\otimes x-4 x\otimes y\otimes x\otimes x\otimes x+y\otimes x\otimes x\otimes x\otimes x\]

Chevalley

a synthesis between the theory of Lie groups and the theory of finite groups

리대수 $\mathfrak{sl}(2)$

$L=\langle E,F,H \rangle$
commutator

\[ [E,F]=H \\ [H,E]=2E \\ [H,F]=-2F \]

observation

from the root system, we can fix $h_{\alpha}$ uniquely for each $\alpha\in \Delta$
we can choose $x_{\alpha}$ so that $[x_{\alpha},x_{-\alpha}]=h_{\alpha}$
the structure constants $n_{\alpha,\beta}$ where

$$[x_{\alpha},x_{\beta}]=n_{\alpha,\beta}x_{\alpha+\beta}$$ is not fixed by the above condition

but if we make another choice $x_{\alpha}'=u_{\alpha}x_{\alpha}$ with $u_{\alpha}u_{\alpha}'=1$, then structure constants satisfy the following property

$$ n_{\alpha,\beta}'n_{-\alpha,-\beta}'=n_{\alpha,\beta}n_{-\alpha,-\beta} $$

lemma

The number $n_{\alpha,\beta}n_{-\alpha,-\beta}$ is given by $-(p+1)^2$ where $p$ is the largest integer $\geq 0$ such that $\beta-p\alpha\in \Delta$

general case

thm

Chevalley bases exist

Q. why is it surprising or non-trivial?
tentative answer : can we check the Jacobi identity?
for example, taking $2x_{\alpha}$ instead of $x_{\alpha}$ still gives integral Lie bracket.

Kostant

universal enveloping algebra의 PBW 기저

\[\{F^kH^lE^m|k,l,m\geq 0\}\]

thm

For each choice of $r_i,s_{\alpha}\geq 0$, form the product in the given order of the elements $$ \binom{h_i}{r_i} $$ and the elements $$ \frac{x_{\alpha}^{s_{\alpha}}}{s_{\alpha}!} $$ for $i=1,\cdots, \ell$ and $\alpha\in \Phi$. Then the resulting collection if a basis for $U_{\mathbb{Z}}$ as a free $\mathbb{Z}$-module

See [H] chapter 26?
for examplem, one can take

\[\{\frac{F^k}{k!}\binom{H}{l}\frac{E^m}{m!}|k,l,m\geq 0\}\]

$\exp(tE)$ and $\exp(tF)$ exist
$\exp(tH)$ does not exist instead $(1+t)^{H}=1+\binom{H}{1}t+\binom{H^2}{2!}t^2+\cdots$ exists

refs

[H] J. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer, (1972).