Borcherds-Kac-Moody Lie algebras

## Borcherds-Kac-Moody Lie algebras

This section reviews definitions and sets notations for Borcherds-Kac-Moody Lie algebras. Standard references are the book of Kac [Kac], the books of Wakimoto [Wak1,Wak2], the survey article of Macdonald [Mac2] and the handwritten notes of Macdonald [Mac2]. Specifically, [Kac, Ch. 2] is a reference for Section 1, [Kac Chs. 3 and 5] for Section 2 and [Kac Ch. 2] for Section 3.

## Constructing a Lie algebra from a matrix

Let $A=\left({a}_{ij}\right)$ be an $n×n$ matrix. Let
 $r=rankA, l=corankA, so that r+l=n.$ 2.1
By rearranging rows and columns we may assume that ${\left({a}_{ij}\right)}_{1\le i,j\le r}$ is nonsingular. Define a $ℂ$-vector space
 2.2
Define ${\alpha }_{1},\dots ,{\alpha }_{n}\in {𝔥}^{*}$ by
 $αihj= aij and αidj= δi,r+j,$ 2.3
and let
 2.4
Let ${c}_{1},\dots ,{h}_{l}\in 𝔥\text{'}$ ba a basis of $𝔠$ so that ${h}_{1},\dots ,{𝔥}_{r},{c}_{1},\dots ,{c}_{l},{d}_{1},\dots ,{d}_{l}$ is another basis of $𝔥$ and define ${\kappa }_{1},\dots ,{\kappa }_{l}\in {𝔥}^{*}$ by
 $κihj=0, κicj=δij, and κidj=0.$ 2.5
Then ${\alpha }_{1},\dots ,{\alpha }_{n},{\kappa }_{1},\dots ,{\kappa }_{l}$ form a basis of ${𝔥}^{*}$. Let $𝔥$ be the Lie algebra given by the generators $𝔥,{e}_{1},\dots {e}_{n},{f}_{1},\dots {f}_{n}$ and relations
 $[h,h']=0, [ei,fj]= δijhi, [h,ei]= αihei, [h,fi]= -αihfi,$ 2.6
for $h,h\text{'}\in 𝔥$ and $1\le i,j\le n$. The Borcherds-Kac-Moody Lie algebra of $A$ is
 2.7
The Lie algebra $𝔞$ is graded by
 2.8
for $h\in 𝔥$. Any ideal of $𝔞$ is $Q$-graded so $𝔤$ is $Q$-graded (see [
Mac2, (1.6)] or [Mac3, p. 81]),
 2.9
The mulpiplicity of a root $\alpha \in R$ is $dim\left({𝔤}_{\alpha }\right)$ and the decomposition of $𝔤$ in (
2.9) is the decomposition of $𝔤$ as an $𝔥$-module (under the adjoint action). If then (see [Mac3, p. 83] or [Kac, §1.3])
 $𝔤=𝔫-⊕𝔥⊕𝔫- and 𝔥=𝔤0, 𝔫+= ⊕ α∈R+ 𝔤α, 𝔫-= ⊕ α∈R- 𝔤α,$ 2.1
where
 $R+=Q+∩ R with Q+= ∑ i=1 n ℤ≥0 αi.$ 2.11
 $𝔤=𝔫-⊕𝔥⊕ 𝔫+=𝔞/𝔯=𝔤'⋊𝔡, 𝔤'== 𝔫-⊕𝔥'⊕𝔫+= [𝔤,𝔤], 𝔤 ‾ ' = 𝔫-⊕ 𝔥 ‾ ' ⊕𝔫+ =𝔤'/𝔠,$ 2.12

and $𝔤\text{'}$ is the universal central extension of $\stackrel{‾}{𝔤}$ (see [
Kac, Exercise 3.14]).

## Cartan matrices, ${𝔰𝔩}_{2}$ subalgebras and the Weyl group

A Cartan matrix is an $n×n$ matrix $A=\left({a}_{ij}\right)$ such that
 2.13
When $A$ is a Cartan matrix the Lie algebra $𝔤$ contains many subalgebras isomorphic to ${𝔰𝔩}_{2}$. For $1\le i\le n$, the elements ${e}_{i}$ and ${f}_{i}$ act locally nilpotently on $𝔤$ (see [
Mac3 p. 85] or [Mac2 (1.19)] or [Kac, Lemma 3.5]),
 $span ei,fi,hi ≅ 𝔰𝔩2, and s ˜ i = expad ei exp-ad fi expad ei$ 2.14
is an automorphism of $𝔤$ (see [
Kac, Lemma 3.8]). Thus $𝔤$ haslots of symmetry.

The simple reflections ${s}_{i}:{𝔥}^{*}\to {𝔥}^{*}$ are given by

 2.15
$\lambda \in {𝔥}^{*}$, $h\in h$, and The Weyl group $W$ is the subgroup of $GL\left({𝔥}^{*}\right)$ (or $GL\left(𝔥\right)$) generated by the simple reflections. The simple reflections on $𝔥$ are reflections in the hyperplanes $𝔥αi = h∈𝔥 ∣ αi=0 , and 𝔠=𝔥W= ⋂ i=1 n 𝔥αi.$ The representations of $W$ on $𝔥$ and ${𝔥}^{*}$ are dual so that The group $W$ is presented by generators ${s}_{i},\dots ,{s}_{n}$ and relations
 2.16
for pairs $i\ne j$ such that ${a}_{ij}{a}_{ji}<4$, where ${}_{ij}=2,3,4,6$ if ${a}_{ij}{a}_{ji}=0,1,2,3,$ respectively (see [
Mac2 (2.12)] or [Kac Proposition 3.13]).

The real roots of $𝔤$ are the elements of the set

 $Rre= ⋃ i=1 n Wαi, and Rim= R\Rre$ 2.17

is the set of imaginary roots of $𝔤$. If $\alpha =w{\alpha }_{i}$ is a real root then there is a subalgebra isomorpic to ${𝔰𝔩}_{2}$ spanned by
 $eα= w ˜ ei, fα= w ˜ fi, and hα = w ˜ hi,$ 2.18
and ${s}_{\alpha }=w{s}_{i}{w}^{-1}$ is a reflection of $W$ acting on $𝔥$ and ${𝔥}^{*}$ by
 $sαλ= λ-λhαα and sαh= h-αhhα, respectively.$ 2.19
 2.2
is a fundamental domain for the action of $W$ on the Tits cone
 2.21
$X={𝔥}_{ℝ}$ if and only if $W$ is finite (see [
Kac Proposition 3.12] and [Mac2, (2.14)]).

## Symmetrizable matrices and invariant forms

A symmetrizable matrix is a matrix $A=\left({a}_{ij}\right)$ such that there exists a diagonal matrix

 2.22
If $⟨,⟩:𝔤×𝔤\to ℂ$ is a $𝔤$-invariant symmetric bilinear form then $⟨ hi,h ⟩= ⟨ ei,fi,h ⟩= -⟨ fi, ei,h ⟩ =αih ⟨ei,fi⟩,$ so that
 2.23
Conversely, if $A$ is a symmetrizable matrix then there is a nondegenerate invariant symmetric bilinear form on $𝔤$ determined by the formulas in (
2.23) (see [Mac2, (3.12)] or [Kac, Theorem 2.2]).

If $A$ is a Cartan matrix and $⟨,⟩:𝔥×𝔥\to ℂ$ is a $W$-invariant symmetric bilinear form then $⟨ hi,h ⟩= -⟨ sihi,h ⟩= -⟨ hi,sih ⟩ =- ⟨hi,h- αihhi ⟩ =- ⟨ hi,h ⟩+ αih ⟨ hi,hi ⟩,$ so that

 2.24
In particular, ${\alpha }_{i}\left({h}_{j}\right){\epsilon }_{i}=⟨{h}_{i},{h}_{j}⟩=⟨⟩={\alpha }_{j}\left({h}_{i}\right){\epsilon }_{j}$ so that $A$ is symmetrizable. Conversely, if $A$ is a symmetrizable Cartan matrix thent there is a nondegenereate $W$-invariant symmetric bilinear form on $𝔥$ determined by the formulas in (2.4) (see [Mac2, (2,26)]).

If ${x}_{\alpha }\in {𝔤}_{\alpha }$, ${y}_{\alpha }\in {𝔤}_{-\alpha }$ then $\left[{x}_{\alpha },{y}_{\alpha }\right]\in \left[{𝔤}_{\alpha },{𝔤}_{-\alpha }\right]\subseteq {𝔤}_{0}=𝔥$ and $⟨h,\left[{x}_{\alpha },{y}_{\alpha }\right]⟩=-⟨\left[{x}_{\alpha },h\right],{y}_{\alpha }⟩=\alpha \left(h\right)⟨{x}_{\alpha },{y}_{\alpha }⟩$, so that

 2.25
determines ${h}_{\alpha }^{\vee }\in 𝔥$. If $\alpha \in {R}_{re}$ and ${e}_{\alpha },{f}_{\alpha },{h}_{\alpha }$ are as in (2.18) then
 ${h}_{\alpha }=\left[{e}_{\alpha },{f}_{\alpha }\right]=⟨{e}_{\alpha },{f}_{\alpha }⟩{h}_{\alpha }^{\vee }\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}⟨{e}_{\alpha },{f}_{\alpha }⟩=\frac{1}{2}⟨{h}_{\alpha },{h}_{\alpha }⟩.$ 2.26
Let
 ${\alpha }^{\vee }=⟨{e}_{\alpha },{f}_{\alpha }⟩\alpha =\frac{1}{2}⟨{h}_{\alpha },{h}_{\alpha }⟩\alpha \phantom{\rule{2em}{0ex}}\text{so that}\phantom{\rule{2em}{0ex}}{\alpha }^{\vee }\left(h\right)=⟨h,{h}_{\alpha }⟩.$ 2.27
Use the vector space isomorphism
 $𝔥 → ∼ 𝔥* h ↦ ⟨h,⋅⟩ hα ↦ α∨ hα∨ ↦ α to identify Q∨= ∑ i=1 n ℤhi and Q*= ∑ i=1 n ℤ αi∨$ 2.28
and write
 2.29

## References

[Bou] N. Bourbaki, Groupes et Algèbres de Lie, Masson, Paris, 1990.

[Cox] H. S. M. Coxeter, The product of the generators of a finite group generated by reflections, Duke Math. J. 18 (1951), 765–782. MR0045109 (13,528d)

[Kac] V. Kac, Infinite-dimensional Lie algebras, Third edition. Cambridge University Press, Cambridge, 1990. MR1104219 (92k:17038)

[Mac2] I. G. Macdonald, Handwritten lecture notes on Kac-Moody algebras, 1983.

[Mac3] I. G. Macdonald, Kac-Moody algebras, in: D. J. Britten, F. W. Lemire and R. V. Moody (Eds.), Lie algebras and related topics (Windsor, Ont., 1984), 69–109, CMS Conf. Proc., 5, Amer. Math. Soc., Providence, RI, 1986. MR0832195 (87j:17021)

[OT] P. Orlik and H. Terao, Arrangements of hyperplanes, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 300. Springer-Verlag, Berlin, 1992. MR1217488 (94e:52014)

[Wak1] M. Wakimoto, Infinite-dimensional Lie algebras, Translated from the 1999 Japanese original by Kenji Iohara. Translations of Mathematical Monographs, 195. Iwanami Series in Modern Mathematics. American Mathematical Society, Providence, RI, 2001. MR1793723 (2001k:17038)

[Wak2] M. Wakimoto, Lectures on infinite-dimensional Lie algebra, World Scientific Publishing Co., Inc., River Edge, NJ, 2001. MR1873994 (2003b:17033)