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

3. Twisted polynomial algebras 31
moreover the algebra is finite over its center hence these algebras are closed
under trace and in fact from 3.4 one can easily deduce a formula for the trace
trx a = 0 if x a is not in the center.
3.5 Maximal order
We have already stressed the importance of orders in a simple algebra, an
important special case is the notion of maximal order which in a non commutative
case replace the notion of an integrally closed domain. First we
summarize some results on maximal order, more details can be found in
[MR87], and at the end of the section we give a relation between maximal
orders and twisted polynomial algebras.
Note. Every twisted polynomial algebra of the form kH[x1, . . .,xn] is an
order.
Given an order R in a central simple algebra D an element a ∈ R is a
non zero divisor in R if and only if it is invertible in D, such an element
is called regular element. Given two orders R1 and R2 let us consider the
following condition:
There exist regular elements a, b ∈ R1 such that R1 ⊂ aR2b. This relation
generates an equivalence of orders and a maximal order is one which is
maximal with respect to this equivalence.
Definition 3.5.1. An order R in a central simple algebra D is called maximal
order if given any central element c ∈ Rand an algebra S with R ⊂ S ⊂ 1
cR we have necessarily that R = S.
We remark an important property of maximal orders:
Property. (i) The center Z of a maximal order R is integrally closed.
(ii) If R is finitely generated algebra over a field k then R is a finite module
over Z.
Corollary 3.5.2. A maximal order in a center simple algebra D of degree
d is closed under the reduced trace and hence it is an algebra in the category
Cd.
Proof. See [DCP93], §4.
We want to discuss now some criteria under which, by degeneration arguments,
we can deduce that an algebra is a maximal order.
The setting that we have chosen is suggested by the work on quantum
groups. We assume to have an algebra R, over some field k, with a commutative
subalgebra A and elements x1, . . .,xk satisfying some special conditions
which we will presently explain.