The Manin Hopf algebra of a Koszul Artin-Schelter regular algebra is quasi-hereditary

TL;DR

This paper demonstrates that the universal Hopf algebra associated with a Koszul Artin-Schelter regular algebra is quasi-hereditary, using a combinatorial monoidal category and Tannaka-Krein formalism to analyze its representations.

Contribution

It introduces an explicit combinatorial rigid monoidal category and shows that the associated universal Hopf algebra is quasi-hereditary and derived equivalent to its representation category.

Findings

aut(A) is quasi-hereditary as a coalgebra

aut(A) is derived equivalent to the representation category of U

Constructs an explicit combinatorial category U for analysis

Abstract

For any Koszul Artin-Schelter regular algebra A, we consider a version of the universal Hopf algebra aut(A) coacting on A, introduced by Manin. To study the representations (i.e. finite dimensional comodules) of this Hopf algebra, we use the Tannaka-Krein formalism. Specifically, we construct an explicit combinatorial rigid monoidal category U, equipped with a functor M to finite dimensional vector spaces such that aut(A)= coend_U(M). Using this pair (U,M) we show that aut(A) is quasi-hereditary as a coalgebra and in addition is derived equivalent to the representation category of U.