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 $A=\left(a_{ij}\right)$ be an $n\times n$ matrix. Let

$$r=rank\left(A\right),l=corank\left(A\right),\text{so that}r+l=n.$$

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

$$𝔥=𝔥\text{'}\oplus 𝔡,\begin{array}{l}𝔥\text{'} \text{has basis} h_1,\dots ,h_n, \text{and}\\ 𝔡 \text{has basis} d_1,\dots ,d_l.\end{array}$$

2.2

Define ${\alpha }_1,\dots ,{\alpha }_n\in {𝔥}^{*}$ by

$${\alpha }_i\left(h_j\right)=a_{ij}\text{and}{\alpha }_i\left(d_j\right)={\delta }_{i,r+j},$$

2.3

and let

$$\overline{𝔥}\text{'}=𝔥\text{'}/𝔠\text{where} 𝔠=\left\{h\in 𝔥\text{'}\mid {\alpha }_i\left(h\right)=0 \text{for all} 1\le i\le n\right\}.$$

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

$${\kappa }_i\left(h_j\right)=0,{\kappa }_i\left(c_j\right)={\delta }_{ij},\text{and}{\kappa }_i\left(d_j\right)=0.$$

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

$$[h,h\text{'}]=0,[e_i,f_j]={\delta }_{ij}{h}_i,[h,e_i]={\alpha }_i\left(h\right)e_i,[h,f_i]=-{\alpha }_i\left(h\right)f_i,$$

2.6

for $h,h\text{'}\in 𝔥$ and $1\le i,j\le n$. The Borcherds-Kac-Moody Lie algebra of $A$ is

$$𝔤=\frac{𝔞}{𝔯},\text{where} 𝔯 \text{is the largest ideal of} 𝔞 \text{such that} 𝔯\cap 𝔥=0.$$

2.7

The Lie algebra $𝔞$ is graded by

$$Q=\sum _{i=1}^n\mathbb{Z}{\alpha }_i,\text{by setting} deg\left(e_i\right)={\alpha }_i, deg\left(f_i\right)=-{\alpha }_i, deg\left(h\right)=0,$$

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 $${𝔫}^{+} \text{is the subalgebra generated by} e_1,\dots ,e_n, \text{and}{𝔫}^{-} \text{is the subalgebra generated by} f_1,\dots ,f_n,$$ then (see [Mac3, p. 83] or [Kac, §1.3])

$$𝔤={𝔫}^{-}\oplus 𝔥\oplus {𝔫}^{-}\text{and}𝔥={𝔤}_0,{𝔫}^{+}=\underset{\alpha \in R^{+}}{\oplus }{𝔤}_{\alpha },{𝔫}^{-}=\underset{\alpha \in R^{-}}{\oplus }{𝔤}_{\alpha },$$

2.10

where

$$R^{+}=Q^{+}\cap \hspace{0.17em}R\text{with}{Q}^{+}=\sum _{i=1}^n{\mathbb{Z}}_{\ge 0}{\alpha }_i.$$

2.11

Let $𝔠$ and $𝔡$ be as in (2.2) and (2.4). Then $$𝔡 \text{acts on} 𝔤\text{'}=[𝔤,𝔤] \text{by derivations,}𝔠=Z\left(𝔤\right)=Z\left(𝔤\text{'}\right),$$

$$\begin{array}{c}𝔤={𝔫}^{-}\oplus 𝔥\oplus {𝔫}^{+}=𝔞/𝔯=𝔤\text{'}⋊𝔡,\\ 𝔤\text{'}=={𝔫}^{-}\oplus 𝔥\text{'}\oplus {𝔫}^{+}=[𝔤,𝔤],\\ \overline{𝔤}\text{'}={𝔫}^{-}\oplus \overline{𝔥}\text{'}\oplus {𝔫}^{+}=𝔤\text{'}/𝔠,\end{array}$$

2.12

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

Cartan matrices, ${𝔰𝔩}_2$ subalgebras and the Weyl group

A Cartan matrix is an $n\times n$ matrix $A=\left(a_{ij}\right)$ such that

$$a_{ij}\in \mathbb{Z},a_{ii}=2,a_{ij}\le 0\text{if} i\ne j,a_{ij}\ne 0\text{if and only if}{a}_{ji}\ne 0.$$

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]),

$$span\left\{e_i,f_i,h_i\right\}\cong {𝔰𝔩}_2,\text{and}{\tilde{s}}_i=exp\left(ad\hspace{0.17em}{e}_i\right)exp\left(-ad\hspace{0.17em}{f}_i\right)exp\left(ad\hspace{0.17em}{e}_i\right)$$

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

$$s_i\lambda =\lambda -\lambda \left(h_i\right){\alpha }_i\text{and}{s}_{}h=h-{\alpha }_i\left(h\right)h_i,\text{for} 1\le i\le n,$$

2.15

$\lambda \in {𝔥}^{*}$, $h\in h$, and $${\tilde{s}}_i{𝔤}_{\alpha }={𝔤}_{s_i\alpha }\text{and}{\tilde{s}}_{i}h=s_{i}h,\text{for} \alpha \in R, h\in 𝔥.$$ 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 $${𝔥}^{{\alpha }_i}=\left\{h\in 𝔥\mid {\alpha }_i=0\right\},\text{and}𝔠={𝔥}^W=\bigcap _{i=1}^n{𝔥}^{{\alpha }_i}.$$ The representations of $W$ on $𝔥$ and ${𝔥}^{*}$ are dual so that $$\lambda \left(wh\right)=\left(w^{-1}\lambda \right)\left(h\right),\text{for} w\in W,\lambda \in {𝔥}^{*},h\in 𝔥.$$ The group $W$ is presented by generators $s_i,\dots ,s_n$ and relations

$$s_i^2=1\text{and}\underset{\underset{m_{ij}\text{factors}}{⏟}}{\phantom{(}{s}_i{s}_j{s}_i{s}_j\dots }=\underset{\underset{m_{ij}\text{factors}}{⏟}}{\phantom{(}{s}_j{s}_i{s}_j{s}_i\dots }$$

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

$$R_{re}=\bigcup _{i=1}^{n}W{\alpha }_i,\text{and}{R}_{im}=R\backslash R_{re}$$

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

$$e_{\alpha }=\tilde{w}{e}_i,f_{\alpha }=\tilde{w}{f}_i,\text{and}{h}_{\alpha }=\tilde{w}{h}_i,$$

2.18

and $s_{\alpha }=w{s}_i{w}^{-1}$ is a reflection of $W$ acting on $𝔥$ and ${𝔥}^{*}$ by

$$s_{\alpha }\lambda =\lambda -\lambda \left(h_{\alpha }\right)\alpha \text{and}{s}_{\alpha }h=h-\alpha \left(h\right)h_{\alpha },\text{respectively.}$$

2.19

Let ${𝔥}_{ℝ}=ℝ-span\left\{h_1,\dots ,h_n,d_1,\dots ,d_l\right\}$. The group $W$ acts on ${𝔥}_{ℝ}$ and the dominant chamber

$$C=\left\{{\lambda }^{\vee }\in {𝔥}_{ℝ}\mid ⟨{\alpha }_i,{\lambda }^{\vee }\ge 0 \text{for all} 1\le i\le n\right\}$$

2.20

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

$X=\bigcup _{w\in W}wC=\left\{{\lambda }^{\vee }\in {𝔥}_{ℝ}\mid ⟨{\alpha }_i,{\lambda }^{\vee }<0 \text{for a finite number of} \alpha \in R^{+}\right\}$

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

$$ℰ=diag\left({\epsilon }_1,\dots ,{\epsilon }_n\right),{\epsilon }_i\in {ℝ}_{\ge 0},\text{such that}Aℰ \text{is symmetric.}$$

2.22

If $⟨,⟩:𝔤\times 𝔤\to ℂ$ is a $𝔤$-invariant symmetric bilinear form then $$⟨h_i,h⟩=⟨\left[e_i,f_i\right],h⟩=-⟨f_i,\left[e_i,h\right]⟩={\alpha }_i\left(h\right)⟨e_i,f_i⟩,$$ so that

$⟨h_i,h⟩={\alpha }_i\left(h\right){\epsilon }_i,\text{where} {\epsilon }_i=⟨e_i,f_i⟩.$

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 $⟨,⟩:𝔥\times 𝔥\to ℂ$ is a $W$-invariant symmetric bilinear form then $$⟨h_i,h⟩=-⟨s_i{h}_i,h⟩=-⟨h_i,s_{i}h⟩=-⟨h_i,h-{\alpha }_i\left(h\right)h_i⟩=-⟨h_i,h⟩+{\alpha }_i\left(h\right)⟨h_i,h_i⟩,$$ so that

$$⟨h_i,h⟩={\alpha }_i\left(h\right){\epsilon }_i,\text{where} {\epsilon }_i=\frac{1}{2}⟨h_i,h_i⟩.$$

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

$\left[x_{\alpha },y_{\alpha }\right]=⟨x_{\alpha },y_{\alpha }⟩h_{\alpha }^{\vee },\text{where} ⟨h,h_{\alpha }^{\vee }⟩=\alpha \left(h\right) \text{for all} h\in 𝔥,$

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

$h_{\alpha }=\left[e_{\alpha },f_{\alpha }\right]=⟨e_{\alpha },f_{\alpha }⟩h_{\alpha }^{\vee }\text{and}⟨e_{\alpha },f_{\alpha }⟩=\frac{1}{2}⟨h_{\alpha },h_{\alpha }⟩.$

2.26

Let

${\alpha }^{\vee }=⟨e_{\alpha },f_{\alpha }⟩\alpha =\frac{1}{2}⟨h_{\alpha },h_{\alpha }⟩\alpha \text{so that}{\alpha }^{\vee }\left(h\right)=⟨h,h_{\alpha }⟩.$

2.27

Use the vector space isomorphism

$$\begin{array}{lcl}𝔥& \stackrel{\sim }{\to }& {𝔥}^{*}\\ h& \mapsto & ⟨h,\cdot ⟩\\ h_{\alpha }& \mapsto & {\alpha }^{\vee }\\ h_{\alpha }^{\vee }& \mapsto & \alpha \end{array}\text{to identify}{Q}^{\vee }=\sum _{i=1}^n\mathbb{Z}{h}_i\text{and}{Q}^{*}=\sum _{i=1}^n\mathbb{Z}{\alpha }_i^{\vee }$$

2.28

and write

$⟨{\lambda }^{\vee },\mu ⟩=\mu \left(h_{\lambda }\right)\text{if} {\lambda }^{\vee }={\lambda }_1{\alpha }_1^{\vee }+\cdots +{\lambda }_n{\alpha }_n^{\vee }\text{and}{h}_{\lambda }={\lambda }_1{h}_1+\cdots +{\lambda }_n{h}_n.$

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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