Borcherds-Kac-Moody Lie algebras

Borcherds-Kac-Moody Lie algebras

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

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 that r+l=n. 2.1
By rearranging rows and columns we may assume that aij1i,jr is nonsingular. Define a -vector space
𝔥=𝔥'𝔡, 𝔥' has basis  h1,,hn,  and 𝔡 has basis  d1,,dl. 2.2
Define α1,,αn𝔥* by
αihj= aij and αidj= δi,r+j, 2.3
and let
𝔥 ' =𝔥'/𝔠 where  𝔠= h𝔥' αih=0   for all   1in . 2.4
Let c1,,hl𝔥' ba a basis of 𝔠 so that h1,,𝔥r, c1,,cl, d1,,dl is another basis of 𝔥 and define κ1,,κl𝔥* by
κihj=0, κicj=δij, and κidj=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 1i,jn. The Borcherds-Kac-Moody Lie algebra of A is
𝔤= 𝔞 𝔯 , where  𝔯  is the largest ideal of  𝔞  such that  𝔯𝔥=0. 2.7
The Lie algebra 𝔞 is graded by
Q= i=1 n αi, by setting  degei= αi,  degfi=- αi,  degh=0, 2.8
for h𝔥. Any ideal of 𝔞 is Q-graded so 𝔤 is Q-graded (see [
Mac2, (1.6)] or [Mac3, p. 81]),
𝔤=𝔤0 αR 𝔤α , where  𝔤α= x𝔤 [h,x]= αhx ,  and R= α α0  and  𝔤α0 is the set of roots of  𝔤. 2.9
The mulpiplicity of a root αR 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,  and 𝔫-  is the subalgebra generated by  f1,,fn, then (see [Mac3, p. 83] or [Kac, §1.3])
𝔤=𝔫-𝔥𝔫- and 𝔥=𝔤0, 𝔫+= αR+ 𝔤α, 𝔫-= αR- 𝔤α, 2.10
R+=Q+R with Q+= i=1 n 0 αi. 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, aii=2, aij0 if ij, aij0 if and only if aji0. 2.13
When A is a Cartan matrix the Lie algebra 𝔤 contains many subalgebras isomorphic to 𝔰𝔩2 . For 1in, the elements ei and fi 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 = expadei exp-adfi expadei 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-λhiαi and sh=h-αi hhi, for 1in, 2.15
λ𝔥*, hh, and s ˜ i 𝔤α= 𝔤siα and s ˜ i h=sih, for αR, h𝔥. 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 = h𝔥 αi=0 , and 𝔠=𝔥W= i=1 n 𝔥αi. The representations of W on 𝔥 and 𝔥* are dual so that λwh= w-1λ h, for wW, λ𝔥*, h𝔥. The group W is presented by generators si,,sn and relations
si2 =1 and ( si sj si sj mij   factors = ( sj si sj si mij   factors 2.16
for pairs ij 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=1 n Wαi, and Rim= 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, and hα = w ˜ hi, 2.18
and sα =wsiw-1 is a reflection of W acting on 𝔥 and 𝔥* by
sαλ= λ-λhαα and sαh= h-αhhα, respectively. 2.19

Let 𝔥= -span h1,,hn, d1,,dl . The group W acts on 𝔥 and the dominant chamber

C= λ 𝔥 αi,λ0  for all  1in 2.20
is a fundamental domain for the action of W on the Tits cone
X= wW wC = λ 𝔥 αi,λ<0  for a finite number of  αR+ 2.21
X=𝔥 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

= diag ε1,,εn , εi0, such that A  is symmetric. 2.22
If ,:𝔤×𝔤 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 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 xα𝔤α, yα𝔤-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 h𝔥, 2.25
determines hα 𝔥 . If αRre and eα,fα,hα are as in (2.18) then
hα= eα,fα = eα,fα hα and eα,fα = 12 hα,hα . 2.26
α= eα,fα α= 12 hα,hα α so that αh= h,hα . 2.27
Use the vector space isomorphism
𝔥 𝔥* h h, hα α hα α to identify Q= i=1 n hi and Q*= i=1 n αi 2.28
and write
λ,μ = μhλ if  λ= λ1 α1 ++ λn αn and hλ= λ1h1 ++ λnhn. 2.29


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