Quantum function algebras from finite-dimensional Nichols algebras

TL;DR

This paper develops a method to construct quantum function algebras from finite-dimensional Nichols algebras, generalizing previous approaches and providing explicit formulas for quantum determinants and antipodes.

Contribution

It introduces a new construction of Hopf algebras from Nichols algebras using the FRT-construction, extending quantum algebra theory.

Findings

Provides explicit formulas for quantum determinants and antipodes.

Constructs Hopf algebras from Nichols algebras in braided categories.

Includes examples from Fomin-Kirillov algebras and symmetric groups.

Abstract

We describe how to find quantum determinants and antipode formulas from braided vector spaces using the FRT-construction and finite-dimensional Nichols algebras. It generalizes the construction of quantum function algebras using quantum grassmanian algebras. Given a finite-dimensional Nichols algebra B, our method provides a Hopf algebra H such that B is a braided Hopf algebra in the category of H-comodules. It also serves as source to produce Hopf algebras generated by cosemisimple subcoalgebras, which are of interest for the generalized lifting method. We give several examples, among them quantum function algebras from Fomin-Kirillov algebras associated with the symmetric groups on three letters.