On quantum groups associated to non-Noetherian regular algebras of dimension 2

Abstract

We investigate homological and ring-theoretic properties of universal quantum linear groups that coact on Artin-Schelter regular algebras A(n) of global dimension 2, especially with central homological codeterminant (or central quantum determinant). As classified by Zhang, the algebras A(n) are connected \N-graded algebras that are finitely generated by n indeterminants of degree 1, subject to one quadratic relation. In the case when the homological codeterminant of the coaction is trivial, we show that the quantum group of interest, defined independently by Manin and by Dubois-Violette and Launer, is Artin-Schelter regular of global dimension 3 and also skew Calabi-Yau (homologically smooth of dimension 3). For central homological codeterminant, we verify that the quantum groups are Noetherian and have finite Gelfand-Kirillov dimension precisely when the corresponding comodule algebra A(n) satisfies these properties, that is, if and only if n=2. We have similar results for arbitrary homological codeterminant if we require that the quantum groups are involutory. We also establish conditions when Hopf quotients of these quantum groups, that also coact on A(n), are cocommutative.