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4.6 The invariant bilinear form

4.6.1 Definition

To proceed, we assume, as announced above, that the Cartan matrix is invertible and symmetrizable since these are the only cases encountered in the billiards. Under these assumptions, an invertible, invariant bilinear form is easily defined on the algebra. We denote by εi the diagonal elements of D,

A = DS, D = diag(ε1,ε2⋅⋅⋅ ,εn). (4.28 )
First, one defines an invertible bilinear form in the dual ∗ 𝔥 of the Cartan subalgebra. This is done by simply setting
(αi|αj ) = Sij (4.29 )
for the simple roots. It follows from Aii = 2 that
2 εi = ------- (4.30 ) (αi|αi)
and thus the Cartan matrix can be written as
(α |α ) Aij = 2--i--j-. (4.31 ) (αi|αi)
It is customary to fix the normalization of S so that the longest roots have (α |α ) = 2 i i. As we shall now see, the definition (4.29View Equation) leads to an invariant bilinear form on the Kac–Moody algebra.

Since the bilinear form (⋅|⋅) is nondegenerate on 𝔥∗, one has an isomorphism μ : 𝔥∗ → 𝔥 defined by

⟨α, μ(γ)⟩ = (α|γ). (4.32 )
This isomorphism induces a bilinear form on the Cartan subalgebra, also denoted by (⋅|⋅). The inverse isomorphism is denoted by ν and is such that
′ ′ ′ ⟨ν(h ),h ⟩ = (h|h ), h,h ∈ 𝔥. (4.33 )
Since the Cartan elements hi ≡ α∨ i obey
⟨α ,α∨⟩ = A , (4.34 ) i j ji
it is clear from the definitions that
∨ ∨ 2μ(αi)- ν(hi) ≡ ν(αi ) = εiαi ⇔ hi ≡ αi = (α |α ), (4.35 ) i i
and thus also
(hi|hj ) = εiεjSij. (4.36 )

The bilinear form (⋅|⋅) can be uniquely extended from the Cartan subalgebra to the entire algebra by requiring that it is invariant, i.e., that it fulfills

([x,y]|z ) = (x |[y,z]) ∀ x,y,z ∈ 𝔤. (4.37 )
For instance, for the ei’s and fi’s one finds
(hi|ej)Akj = (hi|[hk, ej]) = ([hi,hk]|ej) = 0 ⇒ (hi|ej) = 0, (4.38 )
and similarly
(h |f ) = 0. (4.39 ) i j
In the same way we have
Aij(ej|fk) = ([hi,ej]|fk) = (hi|[ej,fk]) = (hi|hj)δjk = Aijεjδjk, (4.40 )
and thus
(ei|fj) = εiδij. (4.41 )
Quite generally, if e α and e γ are root vectors corresponding respectively to the roots α and γ,
[h,e ] = α (h )e , [h,e ] = γ(h)e , α α γ γ

then (eα|eγ) = 0 unless γ = − α. Indeed, one has

α(h )(e |e ) = ([h, e ]|e ) = − (e |[h, e ]) = − γ(h)(e |e ), α γ α γ α γ α γ

and thus

(eα|eγ) = 0 if α + γ ⁄= 0. (4.42 )
It is proven in [116Jump To The Next Citation Point] that the invariance condition on the bilinear form defines it indeed consistently and that it is nondegenerate. Furthermore, one finds the relations
[h,x] = α(h)x, [h,y] = − α(h)y ⇒ [x,y] = (x|y)μ(α). (4.43 )

4.6.2 Real and imaginary roots

Consider the restriction (⋅|⋅)ℝ of the bilinear form to the real vector space 𝔥⋆ ℝ obtained by taking the real span of the simple roots,

⋆ ∑ 𝔥 ℝ = ℝ αi. (4.44 ) i
This defines a scalar product with a definite signature. As we have mentioned, the signature is Euclidean if and only if the algebra is finite-dimensional [116Jump To The Next Citation Point]. In that case, all roots – and not just the simple ones – are spacelike, i.e., such that (α |α ) > 0.

When the algebra is infinite-dimensional, the invariant scalar product does not have Euclidean signature. The spacelike roots are called “real roots”, the non-spacelike ones are called “imaginary roots” [116Jump To The Next Citation Point]. While the real roots are nondegenerate (i.e., the corresponding eigenspaces, called “root spaces”, are one-dimensional), this is not so for imaginary roots. In fact, it is a challenge to understand the degeneracy of imaginary roots for general indefinite Kac–Moody algebras, and, in particular, for Lorentzian Kac–Moody algebras.

Another characteristic feature of real roots, familiar from standard finite-dimensional Lie algebra theory, is that if α is a (real) root, no multiple of α is a root except ±α. This is not so for imaginary roots, where 2α (or other non-trivial multiples of α) can be a root even if α is. We shall provide explicit examples below.

Finally, while there are at most two different root lengths in the finite-dimensional case, this is no longer true even for real roots in the case of infinite-dimensional Kac–Moody algebras11. When all the real roots have the same length, one says that the algebra is “simply-laced”. Note that the imaginary roots (if any) do not have the same length, except in the affine case where they all have length squared equal to zero.

4.6.3 Fundamental weights and the Weyl vector

The fundamental weights {Λi} of the Kac–Moody algebra are vectors in the dual space ∗ 𝔥 of the Cartan subalgebra defined by

∨ ⟨Λi,α j ⟩ = δij. (4.45 )
This implies
(Λi|αj) = δij. (4.46 ) εj

The Weyl vector ∗ ρ ∈ 𝔥 is defined by

(ρ |αj) = 1- (4.47 ) εj
and is thus equal to
ρ = ∑ Λ . (4.48 ) i i

4.6.4 The generalized Casimir operator

From the invariant bilinear form, one can construct a generalized Casimir operator as follows.

We denote the eigenspace associated with α by 𝔤α. This is called the “root space” of α and is defined as

𝔤 = {x ∈ 𝔤 |[x, h] = α(h)x, ∀h ∈ 𝔥 }. (4.49 ) α
A representation of the Kac–Moody algebra is called restricted if for every vector v of the representation subspace V, one has 𝔤α ⋅ v = 0 for all but a finite number of positive roots α.

Let K {eα } be a basis of 𝔤α and let K {e−α } be the basis of 𝔤 −α dual to K {eα } in the B-metric,

(eKα |eL− α) = δKL. (4.50 )
Similarly, let {ui} be a basis of 𝔥 and {ui} the dual basis of 𝔥 with respect to the bilinear form (⋅|⋅),
j j (ui|u ) = δi. (4.51 )
We set
∑ i ∑ ∑ K K Ω = 2μ(ρ) + u ui + 2 e−αeα , (4.52 ) i α∈Δ+ K
where ρ is the Weyl vector. Recall from Section 4.6.1 that μ is an isomorphism from 𝔥⋆ to 𝔥, so, since ρ ∈ 𝔥⋆, the expression μ(ρ) belongs to 𝔥 as required. When acting on any vector of a restricted representation, Ω is well-defined since only a finite number of terms are different from zero.

It is proven in [116Jump To The Next Citation Point] that Ω commutes with all the operators of any restricted representation. For that reason, it is known as the (generalized) Casimir operator. It is quadratic in the generators12.

Note

This definition – and, in particular, the presence of the linear term μ (ρ ) – might seem a bit strange at first sight. To appreciate it, turn to a finite-dimensional simple Lie algebra. In the above notations, the usual expression for the quadratic Casimir operator reads

∑ AB ∑ i ∑ Ω finite = κ TATB = uui + (e−αeα + eαe− α) (4.53 ) A i α∈ Δ+
(without degeneracy index K since the roots are nondegenerate in the finite-dimensional case). Here, AB κ is the Killing metric and {TA} a basis of the Lie algebra. This expression is not “normal-ordered” because there are, in the last term, lowering operators standing on the right. We thus replace the last term by
∑ ∑ ∑ eαe−α = e−αeα + [eα,e− α] α∈Δ+ α∈Δ+ α∈Δ+ ∑ ∑ = e−αeα + μ(α ). (4.54 ) α∈Δ+ α∈Δ+
Using the fact that in a finite-dimensional Lie algebra, ∑ ρ = (1∕2) α∈Δ+ α, (see, e.g., [85Jump To The Next Citation Point]) one sees that the Casimir operator can be rewritten in “normal ordered” form as in Equation (4.52View Equation). The advantage of the normal-ordered form is that it makes sense also for infinite-dimensional Kac–Moody algebras in the case of restricted representations.
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