Quantum function algebras from finite-dimensional Nichols algebras

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 improves the construction of quantum function algebras using quantum grassmanian algebras. Given a finite-dimensional Nichols algebra $\mathfrak{B}$, our method provides a Hopf algebra $H$ such that $\mathfrak{B}$ 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 group on three letters.