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).
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