Differential smoothness of bi-quadratic algebras with PBW basis
Andrés Rubiano, Armando Reyes
TL;DR
This work develops a general framework to determine when bi-quadratic algebras with PBW bases are differentially smooth, extending prior three-variable results to arbitrary numbers of generators. It extends Brzeziński's hom-connection approach to multilinear bi-quadratic settings and proves two main results: a set of explicit sufficient conditions, expressed via vanishing structure constants and commuting automorphisms, that guarantee differential smoothness; and a non-smoothness obstruction that arises when nonzero cross-terms appear. The authors construct an explicit differential calculus with a volume form, and apply these criteria to a broad class of algebras, including well-known examples that are differentially smooth and several not covered by simpler skew extensions, while also validating differential smoothness for the 3-cyclic quantum Weyl family in certain cases. Altogether, the paper provides a practical toolkit for assessing differential smoothness in noncommutative polynomial-type algebras and sets a path for extending these ideas to related algebra families such as double Ore extensions.
Abstract
We investigate the differential smoothness of bi-quadratic algebras with PBW basis.