On quantum groups associated to a pair of preregular forms

Abstract

We define the universal quantum group $\mathcal{H}$ that preserves a pair of Hopf comodule maps, whose underlying vector space maps are preregular forms defined on dual vector spaces. This generalizes the construction of Bichon and Dubois-Violette (2013), where the target of these comodule maps are the ground field. We also recover the quantum groups introduced by Dubois-Violette and Launer (1990), by Takeuchi (1990), by Artin, Schelter, and Tate (1991), and by Mrozinski (2014), via our construction. As a consequence, we obtain an explicit presentation of a universal quantum group that coacts simultaneously on a pair of $N$-Koszul Artin-Schelter regular algebras with arbitrary quantum determinant.

v2: To appear in Journal of Noncommutative Geometry

References in corpus (1)

• The representation theory of noncommutative $\mathcal{O}({\text{GL}}_2)$

Cited by in corpus (1)

• Calabi-Yau property under monoidal Morita-Takeuchi equivalence