Dual reflection groups of graded Ore extensions

Abstract

We prove that the Artin–Schelter Gorenstein and Artin–Schelter regular properties descend from a graded Ore extension to its coefficient algebra, for every twisting automorphism and every skew derivation. We then determine the dual reflection groups of graded Ore extensions for all gradings compatible with the Ore structure. They are exactly the semidirect products of a dual reflection group of the coefficient algebra by the cyclic group generated by the degree of the new variable, meeting it trivially, and a nonzero skew derivation never contributes a new degree. For iterated Ore extensions we characterize the dual reflection groups by an order formula analogous to the theorem of Shephard, Todd and Chevalley, and we prove the binomial conjecture of Buzaglo and Rogalski for these algebras. We show that their covariant rings are complete intersections of growth type, whose Ext-algebras grow like a polynomial ring in as many variables as there are generators of nontrivial degree. This answers a question of Kirkman, Kuzmanovich and Zhang for these algebras. Finally, we describe the smash product as a Cayley quiver with relations, factor its matrix Hilbert series, and show that the extension becomes Hopf–Galois once the Jacobian is inverted.