Borcherds-Kac-Moody Lie algebras

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updates: 29 October 2010

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=aij be an n×n matrix. Let

r=rankA,l=corankA,so thatr+l=n.

2.1

By rearranging rows and columns we may assume that aij1≤i,j≤r is nonsingular. Define a ℂ-vector space

𝔥=𝔥'⊕𝔡,𝔥' has basis h1,…,hn, and𝔡 has basis d1,…,dl.

2.2

Define α1,…,αn∈𝔥* by

αihj=aijandαidj=δi,r+j,

2.3

and let

𝔥‾'=𝔥'/𝔠where 𝔠=h∈𝔥'∣αih=0 for all 1≤i≤n.

2.4

Let c1,…,hl∈𝔥' ba a basis of 𝔠 so that h1,…,𝔥r,c1,…,cl,d1,…,dl is another basis of 𝔥 and define κ1,…,κl∈𝔥* by

κihj=0,κicj=δij,andκidj=0.

2.5

Then α1,…,αn,κ1,…,κl form a basis of 𝔥*. Let 𝔥 be the Lie algebra given by the generators 𝔥,e1,…en,f1,…fn and relations

[h,h']=0,[ei,fj]=δijhi,[h,ei]=αihei,[h,fi]=-αihfi,

2.6

for h,h'∈𝔥 and 1≤i,j≤n. The Borcherds-Kac-Moody Lie algebra of A is

𝔤=𝔞𝔯,where 𝔯 is the largest ideal of 𝔞 such that 𝔯∩𝔥=0.

2.7

The Lie algebra 𝔞 is graded by

Q=∑i=1nℤαi,by setting degei=αi, degfi=-αi, degh=0,

2.8

for h∈𝔥. Any ideal of 𝔞 is Q-graded so 𝔤 is Q-graded (see [Mac2, (1.6)] or [Mac3, p. 81]),

𝔤=𝔤0⊕⊕α∈R𝔤α,where 𝔤α=x∈𝔤∣[h,x]=αhx, andR=α∣α≠0 and 𝔤α≠0is the set of roots of 𝔤.

2.9

The mulpiplicity of a root α∈R is dim𝔤α and the decomposition of 𝔤 in (2.9) is the decomposition of 𝔤 as an 𝔥-module (under the adjoint action). If 𝔫+ is the subalgebra generated by e1,…,en, and𝔫- is the subalgebra generated by f1,…,fn, then (see [Mac3, p. 83] or [Kac, §1.3])

𝔤=𝔫-⊕𝔥⊕𝔫-and𝔥=𝔤0,𝔫+=⊕α∈R+𝔤α,𝔫-=⊕α∈R-𝔤α,

2.10

where

R+=Q+∩ RwithQ+=∑i=1nℤ≥0αi.

2.11

Let 𝔠 and 𝔡 be as in (2.2) and (2.4). Then 𝔡 acts on 𝔤'=[𝔤,𝔤] by derivations,𝔠=Z𝔤=Z𝔤',

𝔤=𝔫-⊕𝔥⊕𝔫+=𝔞/𝔯=𝔤'⋊𝔡,𝔤'==𝔫-⊕𝔥'⊕𝔫+=[𝔤,𝔤],𝔤‾'=𝔫-⊕𝔥‾'⊕𝔫+=𝔤'/𝔠,

2.12

and 𝔤' is the universal central extension of 𝔤‾ (see [Kac, Exercise 3.14]).

Cartan matrices, 𝔰𝔩2 subalgebras and the Weyl group

A Cartan matrix is an n×n matrix A=aij such that

aij∈ℤ,aii=2,aij≤0if i≠j,aij≠0if and only ifaji≠0.

2.13

When A is a Cartan matrix the Lie algebra 𝔤 contains many subalgebras isomorphic to 𝔰𝔩2. For 1≤i≤n, the elements ei and fi act locally nilpotently on 𝔤 (see [Mac3 p. 85] or [Mac2 (1.19)] or [Kac, Lemma 3.5]),

spanei,fi,hi≅𝔰𝔩2,ands˜i=expad eiexp-ad fiexpad ei

2.14

is an automorphism of 𝔤 (see [Kac, Lemma 3.8]). Thus 𝔤 haslots of symmetry.

The simple reflections si:𝔥*→𝔥* are given by

siλ=λ-λhiαiandsh=h-αihhi,for 1≤i≤n,

2.15

λ∈𝔥*, h∈h, and s˜i𝔤α=𝔤siαands˜ih=sih,for α∈R, h∈𝔥. The Weyl group W is the subgroup of GL𝔥* (or GL𝔥) generated by the simple reflections. The simple reflections on 𝔥 are reflections in the hyperplanes 𝔥αi=h∈𝔥∣αi=0,and𝔠=𝔥W=⋂i=1n𝔥αi. The representations of W on 𝔥 and 𝔥* are dual so that λwh=w-1λh,for w∈W,λ∈𝔥*,h∈𝔥. The group W is presented by generators si,…,sn and relations

si2=1and(sisjsisj…⏟mij factors=(sjsisjsi…⏟mij factors

2.16

for pairs i≠j such that aijaji<4, where ij=2,3,4,6 if aijaji=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=1nWαi,andRim=R\Rre

2.17

is the set of imaginary roots of 𝔤. If α=wαi is a real root then there is a subalgebra isomorpic to 𝔰𝔩2 spanned by

eα=w˜ei,fα=w˜fi,andhα=w˜hi,

2.18

and sα=wsiw-1 is a reflection of W acting on 𝔥 and 𝔥* by

sαλ=λ-λhααandsαh=h-αhhα,respectively.

2.19

Let 𝔥ℝ=ℝ-spanh1,…,hn,d1,…,dl. The group W acts on 𝔥ℝ and the dominant chamber

C=λ∨∈𝔥ℝ∣⟨αi,λ∨≥0 for all 1≤i≤n

2.20

is a fundamental domain for the action of W on the Tits cone

X=⋃w∈WwC=λ∨∈𝔥ℝ∣⟨αi,λ∨<0 for a finite number of α∈R+

2.21

X=𝔥ℝ 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=aij such that there exists a diagonal matrix

ℰ=diagε1,…,εn,εi∈ℝ≥0,such thatAℰ is symmetric.

2.22

If ⟨,⟩:𝔤×𝔤→ℂ is a 𝔤-invariant symmetric bilinear form then ⟨hi,h⟩=⟨ei,fi,h⟩=-⟨fi,ei,h⟩=αih⟨ei,fi⟩, so that

⟨hi,h⟩=αihεi,where εi=⟨ei,fi⟩.

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→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

⟨hi,h⟩=αihεi,where εi=12⟨hi,hi⟩.

2.24

In particular, αihjεi=⟨hi,hj⟩=⟨⟩=αjhiεj 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 xα∈𝔤α, yα∈𝔤-α then xα,yα∈𝔤α,𝔤-α⊆𝔤0=𝔥 and ⟨h,xα,yα⟩=-⟨xα,h,yα⟩=αh⟨xα,yα⟩, so that

xα,yα=⟨xα,yα⟩hα∨,where ⟨h,hα∨⟩=αh for all h∈𝔥,

2.25

determines hα∨∈𝔥. If α∈Rre and eα,fα,hα are as in (2.18) then

hα=eα,fα=⟨eα,fα⟩hα∨and⟨eα,fα⟩=12⟨hα,hα⟩.

2.26

Let

α∨=⟨eα,fα⟩α=12⟨hα,hα⟩αso thatα∨h=⟨h,hα⟩.

2.27

Use the vector space isomorphism

𝔥→∼𝔥*h↦⟨h,⋅⟩hα↦α∨hα∨↦αto identifyQ∨=∑i=1nℤhiandQ*=∑i=1nℤαi∨

2.28

and write

⟨λ∨,μ⟩=μhλif λ∨=λ1α1∨+⋯+λnαn∨andhλ=λ1h1+⋯+λnhn.

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)

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