2.2. THE QUANTUM ALGEBRA OF TYPE Ar−1 13

We divide the remaining cases into four subcases according to whether α + αi

and β + αi are elements of Φ or not, and show that eivα,β = 0. We first consider

the case where one of α +

αi, β + αi belongs to Φ. As the argument is the same, we

only treat the case where α + αi ∈ Φ. When α + αi ∈ Φ, we can write [Eαi , Eα] =

cαi,αEαi+α with a non-zero number cαi,α. Thus,

[Lαi , Lα] = eiι(Eα) = ι(eiEα) = ι([Eαi , Eα])

= cαi,αι(Eαi+α) = cαi,αLαi+α.

This means that the element

[[Lαi , Lα] , Lβ] − ι

(

[[Eαi , Eα] , Eβ]

)

is equal to a scalar multiple of vα+αi,β. By the maximality of γ, this element is 0.

Using this we have

eivα,β = [Lα, [Lαi , Lβ]] − ι

(

[Eα, [Eαi , Eβ]]

)

.

If β + αi / ∈ Φ, then β + αi = 0 implies that [Eαi , Eβ] = 0. So

[Lαi , Lβ] = eiι(Eβ) = ι(eiEβ) = ι([Eαi , Eβ]) = 0,

which implies that eivα,β = 0. If β+αi ∈ Φ, then eivα,β is equal to a scalar multiple

of vα,β+αi and the maximality of γ again implies that eivα,β = 0.

If both α + αi,β + αi are not in Φ, then

[Lαi , Lα] = eiι(Eα) = ι(eiEα) = ι([Eαi , Eα]) = 0,

[Lαi , Lβ] = eiι(Eβ) = ι(eiEβ) = ι([Eαi , Eβ]) = 0.

Hence we have eivα,β = 0 in this case also.

Since i is arbitrary and eivα,β = 0, Assertion 3(3) tells us that vα,β is equal to

a scalar multiple of ι(E1n). In particular, we have

γ = α1 + · · · + αn−1

and {α, β} = {α1 + · · · + αk,αk+1 + · · · + αn−1} for some k. However, [Lα, Lβ] is

equal to L1n in this case and so vα,β = 0, which contradicts our choice of γ. Hence,

we have A = ∅ and Assertion 4 follows.

We are now ready to prove Theorem 2.2. We have constructed a map ι : g → U

which satisfies [ι(X), ι(Y )] = ι([X, Y ]). Our next task is to prove the universality

of the pair (U, ι); however, this is obvious because the map φ : U → A is uniquely

determined by the requirements that φ(ei) = ρ(Ei,i+1) etc.

2.2. The quantum algebra of type Ar−1

Based on Theorem 2.2 Drinfeld and Jimbo introduced the quantum algebra

which is obtained as a “deformation”of the enveloping algebra of slr = sl(r, C).

The definition is as follows. We choose Q(v) as a base field since it is not necessary

to assume it to be C(v). The element ti is often denoted by vhi and αj (hi) =

2δij − δi,j+1 − δi+1,j by definition.

Definition 2.5. Let K = Q(v) where v is an indeterminate. The quantum

algebra of type Ar−1 is the unital associative K-algebra Uv(slr) defined by the

following generators and relations.

Generators: ti

±1,ei,fi

(1 ≤ i ≤ r − 1).