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

3. Twisted polynomial algebras 32
Let us first introduce some notations. For an integral vector n :=
(n1, . . .,nk), ni ∈ N we set deg n := n1 + . . . + nk, and xn = x n1
1 . . .xnk k
we call such element a monomial. Furthermore we define on the set of integral
vectors the degree lexicographic ordering, i.e. set n < m if either
deg n < deg m or deg n = deg m but n is less than m in the usual lexicographic
order, in this way Nk becomes an ordered monoid.
We now impose:
1. The monomials xn are a basis of R as a left A module. Let us denote
by
Rn :=
Ax m .
m≤n
2. The subspace Rn gives a structure of filtered algebra with respect to
the ordered monoid N k . Furthermore we restrict the commutation
relations among the elements xi and A
3. xixj = aijxjxi + bij with 0 = aij ∈ k and bij lower than xixj in the
filtration.
4. xia = σi(a)xi + lower term with σi an automorphism of A. Notice
that:
(a) x n x m = λx m x n + lower term, with 0 = λ ∈ k.
(b) The associated graded algebra R is a twisted polynomial ring over
A. In fact the class xi of the xi satisfy
5. A is integrally closed.
xixj = aijxjxi,
xia = σi(a)xi.
6. For every vector n there exists a monomial a = x m such that n ≤ m and
its class a is in the center of R. Let us say for such an m is an almost
central monomial. Such monomial have simple special commutation
rules:
We can finally state our result:
ax m = x m a + lower term, ∀a ∈ A.
x m x s = x m+s + lower term,∀s.
Theorem 3.5.3. Assume that R satisfies hypotheses 1 −6 then R is a maximal
order.