Primes of Height One and a Class of Noetherian Finitely Presented Algebras

Constructions are given of Noetherian maximal orders that are finitely presented algebras over a field K, defined by monomial relations. In order to do this, it is shown that the underlying homogeneous information determines the algebraic structure of the algebra. So, it is natural to consider such algebras as semigroup algebras K[S] and to investigate the structure of the monoid S. The relationship between the prime ideals of the algebra and those of the monoid S is one of the main tools. Results analogous to fundamental facts known for the prime spectrum of algebras graded by a finite group are obtained. This is then applied to characterize a large class of prime Noetherian maximal orders that satisfy a polynomial identity, based on a special class of submonoids of polycyclic-by-finite groups. The main results are illustrated with new constructions of concrete classes of finitely presented algebras of this type. Because of the role of Noetherian algebras in the algebraic approach in noncommutative geometry, new concrete classes of finitely presented algebras recently gained a lot of interest. Via the applications to the solutions of the YangBaxter equation, one such class also widens the interest into other fields, such as mathematical physics. These algebras are finitely generated, say by elements x1, . . . , xn, and they have a presentation defined by ( n 2 ) monomial relations of the form xixj = xkxl so that every word xixj appears at most once in a relation. Such algebras have been extensively studied, for example in [11, 12, 14, 20, 33]. It was shown that they have a rich algebraic structure that resembles that of a polynomial algebra in finitely many commuting generators; in particular, they are prime Noetherian maximal orders. Clearly, these algebras can be considered as semigroup algebras K[S], where S is a monoid defined via a presentation as that of the algebra. It turns out that these algebras are closely related to group algebras since S is a submonoid of a torsion-free abelian-by-finite group. Research partially supported by the Onderzoeksraad of Vrije Universiteit Brussel, Fonds voor Wetenschappelijk Onderzoek (Flanders), Flemish-Polish bilateral agreement BIL2005/VUB/06 and a research grant of the Polish Ministry of Science and Higher Education. Research funded by a Ph.D grant of the Institute for the Promotion of Innovation through Science and Technology in Flanders (IWT-Vlaanderen). 2000 Mathematics Subject Classification. Primary 16P40, 16H05, 16S36; Secondary 20M25, 16S15.