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Noetherian semigroup algebras and prime maximal orders

vrijdag, 25 april, 2008 - 16:00
Campus: Brussels Humanities, Sciences & Engineering campus
Faculteit: Science and Bio-engineering Sciences
D
0.07
Isabel Goffa
doctoraatsverdediging

Let S be a semigroup and K be a field. A K-space K[S], with basis S and
with multiplication extending, in a natural way, the operation on S, is called
a semigroup algebra. It remains an open problem to characterize semigroup
algebras that are a prime Noetherian maximal order.

In this thesis, we give an answer to the problem for a large class of cancellative
semigroups and we illustrate these results with several examples of
concrete classes of Noetherian maximal orders. Indeed, we find necessary and
sufficient conditions for a prime Noetherian algebra K[S] of a submonoid S
of a polycyclic-by-finite group to be a maximal order. Under an invariance
condition on the minimal primes, our result is entirely in terms of the monoid
S and, in order to prove it, we describe the height one prime ideals of K[S].
Recall that it is conjectured that polycyclic-by-finite groups G are the only
groups having a Noetherian group algebra and K.A. Brown characterized when
these group algebras K[G] are a prime Noetherian maximal order.

In case K[S] also satisfies a polynomial identity, this means in case S is
a submonoid of a finitely generated abelian-by-finite group, we show that the
invariance condition on the minimal primes of S is necessary for K[S] to be a
prime Noetherian maximal order. Furthermore, we establish a general method
for constructing non-abelian maximal order semigroup algebras of finitely generated
submonoids of abelian-by-finite groups, starting from abelian maximal
orders. To obtain concrete constructions, we thus also need to deal with
abelian finitely presented monoids A. If A has a presentation with one or two
defining relations, we determine necessary and sufficient conditions for K[A]
to be a domain that is a maximal order. The description is only in terms
of the defining relations. Furthermore, we compute the class groups of such
semigroup algebras.

In the appendix, we briefly explain applications of maximal orders in spacetime
coding. These applications softly point out that maximal orders not only
might be interesting for experts in algebra, but also for specialists in coding
theory.