Degree of Parabolic Quantum Groups - Dipartimento di Matematica ...

4. General Theory 48
4.5 Degree of Uǫ(b)
Recall that, we have Uq(b) = U ≥0 = U + U 0 ⊂ Uq(g). We begin to give a
more useful construction of Uq(b). We have seen at the end of section 4.1.1
that for all ω ∈ W, we can construct two twisted derivation algebras U ω and
B ω .
Lemma 4.5.1. Let ω0 ∈ W be the longest element, then Uq(b) = B ω0 .
Proof. Follows from the definitions of Uq(b) and B ω0 .
So, Uq(b) is a twisted derivation algebra. Let ǫ be a primitive l th root of
1 such that l > di for all i, we may consider the specialized algebra
Uǫ(b) = Uq(b)/(q − ǫ) ⊂ Uǫ(g).
Proposition 4.5.2. (i) The monomials
E k1 . . .Ekn β1 βN Ks1 1 . . .Ksn n
for (k1, . . .,kN) ∈ (Z + ) N and (s1, . . .,sn) ∈ Z n form a C basis of Uǫ(b)
(ii) The L.S. relations holds in Uǫ(b).
Proof. See [DCKP95] or [DCP93] §10.
Using previous proposition and proposition 4.1.5, we have that Uǫ(b) is
a quasi derivation algebra with relations of type in example 3.2.8, so we can
consider the associated quasi polynomial algebra Uǫ(b). We can then apply
the theorem 3.2.9. We have
Theorem 4.5.3. The algebras Uǫ(b) and Uǫ(b) have the same degree.
Note. We have that Uǫ(b) ⊂ Gr Uǫ(g).
So the algebra Uǫ(b) is a twisted polynomial algebra where the commutation
relation are of type
xixj = ǫ hij xjxi,
in order to compute is degree d is necessary to identify and study the corresponding
matrix H = (hij) since, according to proposition 3.4.3, d2 is equal
to the number from elements of the image of H modulo l. Let us explicit
the matrix H.
Let xm denote the class of Eβm for m = 1, . . .,N, then from relations
4.9, we have
xixj = ǫ (βi|βj) xjxi, if 0 < i < j