Semigroup algebras and Noetherian maximal orders: a survey.
This is a survey of recent results on semigroup algebras \(K[S]\), of monoids \(S\) over a field \(K\), that are prime Noetherian maximal orders. First, a characterization of submonoids \(S\) of polycyclic-by-finite groups that lead to such algebras \(K[S]\) is presented. Then, certain classes of commutative maximal orders (or equivalently, integrally closed Noetherian domains) are discussed in terms of a finite presentation. Algebras related to the set theoretic solutions of the Yang-Baxter equation are then presented, which can be described as semigroup algebras of certain noncommutative monoids defined by quadratic relations, called monoids of \(I\)-type. Results concerning a generalization of the latter class to the so-called monoids of \(IG\)-type and their algebras are also discussed. An extensive bibliography is provided.
submonoids of polycyclic-by-finite groups
scientific article; zbMATH DE number 5644665
scientific article