Kac-Moody Lie algebras

Arun Ram
Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA

Last updates: 24 June 2007

Kac-Moody Lie algebras

Let $A = (a_{i j})$ be an $n \times n$ matrix. Let

r = rank (A), ℓ = corank (A), so that r + ℓ = n .

By rearranging rows and columns we may assume that

(a_{i j})_{1 \leq i, j \leq r}

is nonsingular. Define a vector space

𝔥 = 𝔥^{'} \oplus 𝔡 where \begin{array}{l} 𝔥^{'} has basis h_{1}, \dots, h_{n}, and \\ 𝔡 has basis d_{1}, \dots, d_{ℓ} . \end{array}

Define

α_{1}, \dots, α_{n} \in 𝔥^{*}

α_{i} (h_{j}) = a_{i j} and α_{i} (d_{j}) = δ_{i, r + j}

and let

{\overline{𝔥}}^{'} = \frac{𝔥^{'}}{𝔠} where 𝔠 = {h \in 𝔥^{'} ∣ α_{i} (h) = 0 for all 1 \leq i, j \leq n}

Let

c_{1}, \dots, c_{ℓ} \in 𝔥^{'}

be 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

ω_{1}, \dots, ω_{ℓ} \in 𝔥^{*}

ω_{i} (h_{j}) = 0, for 1 \leq j \leq r, ω_{i} (c_{j}) = δ_{ij}, ω_{i} (d_{j}) = 0

PUT THIS IN A MATRIX.

Let $𝔞$ be the Lie algebra given by generators $𝔥, e_{1}, \dots, e_{n}, f_{1}, \dots, f_{n}$ and relations

[h, h^{'}] = 0, [e_{i}, f_{j}] = δ_{i j} h_{i}, [h, e_{i}] = α_{i} (h) e_{i}, [h, f_{i}] = - α_{i} (h) f_{i},

for

h, h^{'} \in 𝔥

. The Borcherds-Kac-Moody Lie algebra of

A

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

The Lie algebra $𝔞$ is graded by

Q = \sum_{i \in I} ℤ α_{i} by setting \deg (e_{i}) = α_{i}, \deg (f_{i}) = - α_{i}, \deg (h) = 0

for

h \in 𝔥

. Any ideal of

𝔞

Q

-graded and so

𝔤

Q

-graded,

𝔤 = ⨁_{α \in R} 𝔤_{α}, where 𝔤_{0} = 𝔥, and R = {α ∣ 𝔤_{α} \neq 0}

is the set of roots of

𝔤

. The multiplicity of a root

α

\dim (𝔤_{α})

and the decomposition of

𝔤

in (1.3???) is the decomposition of

𝔤

as an

𝔥

-module (under the adjoint action). If

$𝔫^{+}$ is the subalgebra of $𝔤$ generated by $e_{1}, \dots, e_{n}$ , and
$𝔫^{-}$ is the subalgebra of $𝔤$ generated by $f_{1}, \dots, f_{n}$ ,

then

𝔤 = 𝔫^{-} \oplus 𝔥 \oplus 𝔫^{+} and 𝔥 = 𝔤_{0}, 𝔫^{+} = ⨁_{α \in R^{+}} 𝔤_{α}, 𝔫^{-} = ⨁_{α \in R^{+}} 𝔤_{- α},

where

R^{+} = Q^{+} \cap R with Q^{+} = \sum_{i = 1}^{n} ℤ_{\geq 0} α_{i} .

Let $𝔠$ and $𝔡$ be as in (???) and let

𝔤^{'} = [𝔤, 𝔤] . Then 𝔠 = Z (𝔤) = Z (𝔤^{'})

and

𝔡

acts on

𝔤^{'}

by derivations. Then

\begin{array}{l} 𝔤 = 𝔤 (A) / 𝔯 = 𝔤^{'} ⋊ 𝔡 = 𝔫^{-} \oplus 𝔥 \oplus 𝔫^{+}, \\ 𝔤^{'} = [𝔤, 𝔤] = 𝔫^{-} \oplus 𝔥^{'} \oplus 𝔫^{+}, \\ {\overline{𝔤}}^{'} = 𝔤^{'} / 𝔠 = 𝔫^{-} \oplus {\overline{𝔥}}^{'} \oplus 𝔫^{+}, \end{array}

and

𝔤^{'}

is the universal central extension of

{\overline{𝔤}}^{'}

(see [Kac, Ex. 3.14]).

Cartan matrices and the Weyl group

A Cartan matrix is an $n \times n$ matrix $A = (a_{i j})$ such that

a_{i j} \in ℤ, a_{i i} = 2, a_{i j} \leq 0 if i \neq j, a_{i j} \neq 0 if and only if a_{j i} \neq 0

When

A

is a Cartan matrix the Lie algebra

𝔤

contains many subalgebras isomorphic to

{𝔰𝔩}_{2}

. For

1 \leq i \leq n

, the elements

e_{i}

and

f_{i}

act locally nilpotently on

𝔤

span {e_{i}, f_{i}, h_{i}} ≃ {𝔰𝔩}_{2} and \tilde{s_{i}} = \exp (ad e_{i}) \exp (- ad f_{i}) \exp (ad e_{i})

are automorphisms of

𝔤

. Thus

𝔤

has lots of symmetry.

The simple reflections are $s_{i} : 𝔥^{*} \to 𝔥^{*}$ and $s_{i} : 𝔥 \to 𝔥$ by

s_{i} λ = λ - λ (h_{i}) α_{i}, and s_{i} h = h - h (α_{i}) h_{i}, for 1 \leq i \leq n,

λ \in 𝔥^{*}

h \in 𝔥

, and

\tilde{s_{i}} 𝔤_{α} = 𝔤_{s_{i} α} and \tilde{s_{i}} h = s_{i} h, for α \in R, h \in 𝔥 .

The Weyl group

W

is the subgroup of

GL (𝔥^{*})

generated by the simple reflections. The simple reflections are reflections in the hyperplanes

𝔥^{s_{i}} = {h \in 𝔥 ∣ α_{i} (h) = 0} and 𝔠 = 𝔥^{W} = ⋂_{i = 1}^{n} 𝔥^{s_{i}} .

The representations of

W

𝔥

and

𝔥^{*}

are dual so that

λ (w h) = (w^{- 1} λ) (h), for w \in W, λ \in 𝔥^{*}, h \in 𝔥 .

The group $W$ is presented by

generators s_{1}, \dots, s_{n} and relations s_{i}^{2} = 1 and \underset{\underset{m_{ij} factors}{⏟}}{s_{i} s_{j} s_{i} \dots} = \underset{\underset{m_{ij} factors}{⏟}}{s_{j} s_{i} s_{j} \dots},

where

m_{ij} = a_{ij} a_{ji}

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

R_{re} = \cup_{i = 1}^{n} W α_{i} and R_{im} = R ∖ R_{re}

is the set of imaginary roots of

𝔤

. If

α = w α_{i}

is a real root then

x_{α} = \tilde{w} e_{i}

y_{α} = \tilde{w} f_{i}, h_{α} = \tilde{w} h_{i}

, generate a subalgebra isomorphic to

{𝔰𝔩}_{2}

α (h_{α}) = 2

, and the reflections

s_{α} = w s_{i} w^{- 1}

are the elements of

W

acting on

𝔥

and

𝔥^{*}

s_{α} λ = λ - λ (h_{α}) α, and s_{α} h = h - α (h) h_{α}, respectively.

Symmetrizable matrices and invariant forms

A symmetrizable matrix is a matrix $A = (a_{i j})$ such that there exists a diagonal matrix $ℰ = diag (ε_{1}, \dots, ε_{n})$ with $ε_{i} \in ℝ_{> 0}$ such that $A ℰ$ is symmetric.

If $⟨, ⟩ : 𝔥 \times 𝔥 \to ℂ$ is a $W$ -invariant symmetric bilinear form then $⟨ h_{i}, h ⟩ = - ⟨ s_{i} h_{i}, h ⟩ = - ⟨ h_{i}, s_{i} h ⟩ = - ⟨ h_{i}, h - α_{i} (h) h_{i} ⟩ = - ⟨ h_{i}, h ⟩ + α_{i} (h) ⟨ h_{i}, h_{i} ⟩$ . So

⟨ h_{i}, h ⟩ = α_{i} (h) ε_{i}, where ε_{i} = \frac{1}{2} ⟨ h_{i}, h_{i} ⟩ .

In particular,

α_{i} (h_{j}) ε_{i} = ⟨ h_{i}, h_{j} ⟩ = ⟨ h_{j}, h_{i} ⟩ = α_{j} (h_{i}) ε_{j}

so that

A

is symmetrizable.

If $⟨, ⟩ : 𝔤 \times 𝔤 \to ℂ$ is an invariant symmetric bilinear form then $⟨ h_{i}, h ⟩ = - ⟨ [e_{i}, f_{i}], h ⟩ = - ⟨ f_{i}, [e_{i} h] ⟩ = ⟨ h_{i}, α_{i} (h) e_{i} ⟩ = α_{i} (h) ⟨ e_{i}, f_{i} ⟩$ . So

⟨ h_{i}, h ⟩ = α_{i} (h) ε_{i}, where ε_{i} = ⟨ e_{i}, f_{i} ⟩ .

x_{α} \in 𝔤_{α}

and

x_{β} \in 𝔤_{β}

then

α (h) = ⟨ x_{α}, x_{β} ⟩ = ⟨ [h, x_{α}], x_{β} ⟩ = - ⟨ x_{α}, [h, x_{β}] ⟩ = - β (h) ⟨ x_{α}, x_{β} ⟩

so that

⟨ x_{α}, x_{β} ⟩ = 0, unless α = - β .

x_{α} \in 𝔤_{α}

y_{α} \in 𝔤_{- α}

then

[x_{α}, y_{α}] \subseteq [𝔤_{α}, 𝔤_{- α}] \subseteq 𝔤_{0} = 𝔥_{0}

and

⟨ h, [x_{α}, y_{α}] ⟩ = - ⟨ [x_{α}, h], y_{α} ⟩ = α (h) ⟨ x_{α}, y_{α} ⟩

. So

[x_{α}, y_{α}] = ⟨ x_{α}, y_{α} ⟩ h_{α}^{\lor}, where h_{α}^{\lor} \in 𝔥 is given by ⟨ h, h_{α}^{\lor} ⟩ = α (h),

for all

h \in 𝔥

. The element

h_{α}^{\lor}

is not always equal to the element

h_{α}

appearing in (???). Even for a simple root,

h_{α_{i}}^{\lor} = \frac{2}{⟨ h_{i}, h_{i} ⟩} h_{i}

, by (???).

If $A$ is a generalised Cartan matrix the elements

$(ad e_{i})^{1 - a_{i j}} (e_{j})$ and $(ad f_{i})^{1 - a_{i j}} (f_{j})$ are in $𝔯$ and it is known that if $A$ is symmetrizable these elements generate $𝔯$ .

Loop Lie algebras

Let $𝔤_{0}$ be a Lie algebra with an invariant symmetric bilinear form $⟨, ⟩_{0}$ . Let $R$ be a commutative algebra. Define

{\overline{𝔤}}^{'} = R \otimes_{ℂ} 𝔤_{0} with [r_{1} x_{1}, r_{2} x_{2}] = r_{1} r_{2} [x_{1}, x_{2}]_{0} .

Then

der ({\overline{𝔤}}^{'}) = der (R) \otimes {id}_{𝔤_{0}} + ad ({\overline{𝔤}}^{'}) .

Let

γ \in R^{*}

, the dual of

R

, and define an invariant symmetric bilinear form on

{\overline{𝔤}}^{'}

$⟨ r_{1} x_{1}, r_{2} x_{2} ⟩ = γ (r_{1} r_{2}) ⟨ x_{1}, x_{2} ⟩_{0} .$

If $d \in der (R)$ such that $γ \circ d = 0$ then

$⟨ d (r_{1} x_{1}), r_{2} x_{2} ⟩ = - ⟨ r_{1} x_{1}, d (r_{2} x_{2}) ⟩ and ψ (r_{1} x_{1}, r_{2} x_{2}) = ⟨ d (r_{1} x_{1}), r_{2} x_{2} ⟩$

is a 2-cocycle on ${\overline{𝔤}}^{'}$ which determines a central extension

$𝔤^{'} = {\overline{𝔤}}^{'} \oplus ℂ c with bracket [r_{1} x_{1}, r_{2} x_{2}] = r_{1} r_{2} [x_{1}, x_{2}]_{0} + ψ (r_{1} x_{1}, r_{2} x_{2}) c .$

The derivation $d$ extends to a derivation of ${\overline{𝔤}}^{'}$ and the semidirect product

$𝔤 = 𝔤^{'} \oplus ℂ d = {\overline{𝔤}}^{'} \oplus ℂ c \oplus ℂ d has [d, r x] = d (r x) = d (r) x, and [d, c] = 0 .$

We will apply this construction in the case

$R = ℂ [t, t^{- 1}] where der (ℂ [t, t^{- 1}]) = ⨁_{j \in ℤ} ℂ d_{j}$

with

$d_{j} = t^{j + 1} \frac{d}{d t} and [d_{i}, d_{j}] = (j - i) d_{i + j} .$

Use

$γ (t^{k}) = δ_{k, - 1} and d = \frac{d}{d t}$

to construct the 2-cocycle $ψ$ . In this case every 2-cocycle on ${\overline{𝔤}}^{'}$ is equivalent to $κ ψ$ for some $κ \in ℂ$ and $𝔤^{'}$ is the universal central extension of ${\overline{𝔤}}^{'}$ (see [Kac, Ex. 7.8]).

Let

𝔤 = 𝔤_{0} [t, t^{- 1}] \oplus ℂ c \oplus ℂ d, 𝔤^{'} = 𝔤_{0} [t, t^{- 1}] \oplus ℂ c, {\overline{𝔤}}^{'} = 𝔤_{0} [t, t^{- 1}] = \frac{𝔤^{'}}{ℂ c},

where the bracket on

𝔤

is given by

[t^{m} x, t^{n} y] = t^{m + n} [x, y]_{0} + δ_{m + n, 0} m ⟨ x, y ⟩_{0} c, c is central in 𝔤, and [d, t^{m} x] = m t^{m - 1} x,

for

x \in 𝔤_{0}

m, n \in ℤ

. The invariant symmetric bilinear form on

𝔤

is given by

$⟨ c, d ⟩ = 1, ⟨ t^{m} x, t^{n} y ⟩ = {\begin{cases} ⟨ x, y ⟩_{0}, & if m + n = 0, \\ 0, & otherwise, \end{cases}$
$⟨ c, t^{m} y ⟩ = ⟨ d, t^{n} y ⟩ = 0 and ⟨ c, c ⟩ = ⟨ d, d ⟩ = 0,$

for

x, y \in 𝔤_{0}

m, n \in ℤ

. Then

𝔤 = 𝔤^{'} ⋉ ℂ d

where the derivation

d

acts on

𝔤^{'}

d = t \frac{d}{d t} so that d (c) = 0 and d (t^{m} x) = m t^{m} x,

for

x \in 𝔤_{0}

m \in ℤ

. Then

𝔤^{'}

is the universal central extension of

{\overline{𝔤}}^{'}

via the 2-cocycle on

{\overline{𝔤}}^{'}

given by

ψ (t^{m} x, t^{n} y) = ⟨ d (t^{m} x), t^{n} y ⟩ = ⟨ m t^{m} x, t^{n} y ⟩ = {\begin{cases} m ⟨ x, y ⟩_{0}, & if m + n = 0, \\ 0, & otherwise . \end{cases}

Let

𝔥 = 𝔥_{0} \oplus ℂ c \oplus ℂ d, 𝔥^{'} = 𝔥_{0} \oplus ℂ c, {\overline{𝔥}}^{'} = 𝔥_{0} .

Let

{h_{0}, \dots, h_{n}, c, d}

be a basis of

𝔥

and

{ω_{1}, \dots, ω_{n}, δ, ω_{0}}

the dual basis of

𝔥^{*}

so that

δ (d) = 1 and ω_{0} (c) = 1 .

As an

𝔥

-module

𝔤 = (\underset{\underset{k \in ℤ}{α \in R}}{\oplus} 𝔤_{α + k δ}) \oplus (\underset{\underset{k \neq 0}{k \in ℤ}}{\oplus} 𝔤_{k δ}) \oplus 𝔥_{0} \oplus ℂ c \oplus ℂ d,

where

𝔤_{α + k δ} = t^{k} 𝔤_{α} and 𝔤_{k δ} = t^{k} 𝔥_{0} .

If $α \in R_{re}$ with $α = w α_{i}$ then

⟨ e_{α}, f_{α} ⟩_{0} = ⟨ \tilde{w} e_{i}, \tilde{w} f_{i} ⟩_{0} = ⟨ e_{i}, f_{i} ⟩_{0} and span {t^{- k} e_{α}, t^{k} f_{α}, h_{α} - k c ⟨ e_{α}, f_{α} ⟩_{0}} ≃ 𝔰 𝔩_{2} .

The elements

t^{- k} e_{α}

and

t^{k} f_{α}

act locally nilpotently on

𝔤

because

e_{α}

and

f_{α}

act locally nilpotently on

𝔤_{0}

. Let

h_{α - k δ} = h_{α} - k c ⟨ e_{α}, f_{α} ⟩_{0}

and define

s_{α - k δ} : 𝔥^{*} \to 𝔥^{*}

and

s_{α - k δ} : 𝔥 \to 𝔥

s_{α - k δ} λ = λ - λ (h_{α - k δ}) (α - k δ) and s_{α - k δ} h = h - (α - k δ) (h) h_{α - k δ},

for

λ \in 𝔥^{*}

and

h \in 𝔥

. Then

{\tilde{s}}_{α - k δ} = \exp ({ad}_{t^{- k} e_{α}}) \exp (- {ad}_{t^{k} f_{α}}) \exp ({ad}_{t^{- k} e_{α}})

is a well defined automorphism of

𝔤

and

{\tilde{s}}_{α - k δ} 𝔤_{β} = 𝔤_{s_{α - k δ} β} and {\tilde{s}}_{α - k δ} h = s_{α - k δ} h

for

h \in 𝔥

and

β \in \tilde{R}

, where

\tilde{R} = (R + ℤ δ) ⋃ ℤ_{\neq 0} δ .

The affine Lie algebra

Let $𝔤$ be a finite dimensional complex semisimple Lie algebra with Killing form $⟨, ⟩ : 𝔤 \times 𝔤 \to ℂ$ . Viewing elements of the loop Lie algebra

$𝔤 [t, t^{- 1}] = 𝔤 \otimes ℂ [t, t^{- 1}]$ with bracket $[x t^{m}, y t^{n}] = [x, y] t^{m + n}$

as functions in $t$ identifies $𝔤 [t, t^{- 1}]$ with the space of maps $S^{1} \to 𝔤$ , where $S^{1} = {e^{i θ} ∣ 0 \leq θ < 2 π}$ . Since the space $ℂ c$ dual to the space of $𝔤$ -invariant bilinear forms on $𝔤$ is one dimensional the universal central extension of $𝔤 [t, t^{- 1}]$ is

$𝔤 [t, t^{- 1}] \oplus ℂ c$ with $[x t^{m}, y t^{n}] = [x, y] t^{m + n} + m δ_{m, - n} ⟨ x, y ⟩ c$ ,

where $c$ is central. The derivation $d : 𝔤 [t, t^{- 1}] \to 𝔤 [t, t^{- 1}]$ given by $d = t \frac{d}{d t}$ extends to $𝔤 [t, t^{- 1}] \oplus ℂ c$ by $d (c) = 0$ and adjoining this derivation gives the Lie algebra

$\tilde{𝔤} = 𝔤 [t, t^{- 1}] \oplus ℂ c \oplus ℂ d$ with $[d, x t^{m}] = d (x t^{m}) = m x t^{m - 1}, [d, c] = 0$ .

Let $ϕ$ be the highest root of $𝔤$ , normalise the scalar product on $𝔤$ so that $⟨ ϕ, ϕ ⟩ = 2 ? ? ? ?$ and let $x_{ϕ} \in 𝔤, x_{- ϕ} \in 𝔤_{- ϕ}$ such that $⟨ x_{ϕ}, x_{- ϕ} ⟩ = 1$ . Let

$e_{0} = - x_{- ϕ} t, f_{0} = x_{ϕ} t^{- 1}$ and $h_{0} = [e_{0}, f_{0}] = [t x_{- ϕ}, t^{- 1} x_{ϕ}] = - h_{ϕ^{\lor}} + c$ .

Let

$\tilde{𝔥} = 𝔥 \oplus ℂ c \oplus ℂ d$ , and let $\tilde{𝔥^{*}} = 𝔥^{*} \oplus ℂ γ \oplus ℂ δ$ ,

with $α (c) = α (d) = 0$ , for $α \in 𝔥^{*}$ , and $δ (𝔥 \oplus ℂ c) = 0$ , $δ (d) = 1$ .

Let

$α_{0} = α - ϕ$ , $α_{0}^{\lor} = c - ϕ^{\lor}$ , and $A^{(1)} = (⟨ α_{i}, α_{j}^{\lor} ⟩)_{0 < i, j < ℓ}$ .

Then $e_{0}, \dots, e_{n}, h_{0}, \dots, h_{n}, f_{1}, \dots, f_{n}$ are generators of $\tilde{𝔤}$ as an affine Kac-Moody Lie algebra with Cartan matrix $A^{(1)}$ and minimal realization

$(\tilde{𝔥}, (α_{0}^{\lor}, \dots, α_{n}^{\lor}), (α_{0}, \dots, α_{n}))$ .

Note that $\dim (\tilde{𝔥}) = n + 2 = 2 (n + 1) - rank (A)$ since $rank (A) = n$ . The center of $\tilde{𝔤}$ is $ℂ c$ , the Cartan subalgebra of $\tilde{𝔤}$ is

$\tilde{𝔥} = 𝔥 \oplus ℂ c \oplus ℂ d$ and $𝔤 [t, t^{- 1}] ≃ \frac{{\tilde{𝔤}}^{'}}{ℂ c}$ , where ${\tilde{𝔤}}^{'} = 𝔤 [t, t^{- 1}] \oplus ℂ c$

is the derived algebra of $\tilde{𝔤}$ . Define $δ \in {\tilde{𝔥}}^{*}$ by

$δ (d) = 1$ , $δ (c) = 0$ , $δ (h) = 0$ , for $h \in 𝔥$ .

The real and imaginary roots of $\tilde{𝔤}$ are

${\tilde{R}}_{re} = {α + r δ ∣ α \in R, r \in ℤ}, {\tilde{R}}_{im} = {r δ ∣ r \in ℤ, r \neq 0}$ ,

respectively, where each imaginary root has multiplicity $n = rank (𝔤)$ .

References

[Gao] Y. Gao, Central extensions of nonsymmetrizable Kac-Moody algebras over commutative rings Proc. Amer. Math. Soc. 121 (1994), no. 1, 67-76. MR1185261 (94g:17046)

[Had] A. Haddi, Détermination des extensions centrales des algèbres de Kac-Moody C. R. Acad. Sci. Paris Sér. I Math 306 (1988), no. 16, 691--694. MR0944412 (89i:17018)

[Kac] V. Kac, Infinite dimensional Lie algebras, Second edition, Cambridge University Press, 1985.