Stacks related to classical generalized Weyl algebras

Noncommutative rings arise naturally in many contexts. Given a commutative ring R and a nonabelian group G, the group ring R[G] is a noncommutative ring. The n × n matrices with entries in C, or more generally, the linear transformations of a vector space under composition form a noncommutative ring. Noncommutative rings also arise as differential operators. The ring of differential operators on k[t] generated by multiplication by t and differentiation by t is isomorphic to the first Weyl algebra, A1 := k〈x, y〉/(xy − yx − 1), a celebrated noncommutative ring. From the perspective of physics, the Weyl algebra captures the fact that in quantum mechanics, the position and momentum operators do not commute.