Universal quantum (semi)groups and Hopf envelopes

TL;DR

This paper shows that under certain conditions, the Hopf envelope of a quantum algebra can be obtained by a simple localization, generalizing previous results and providing a universal construction for related algebraic structures.

Contribution

It proves that the Hopf envelope of the FRT construction is a localization by the quantum determinant, removing unnecessary hypotheses from earlier work.

Findings

The Hopf envelope is the localization of the FRT algebra by the quantum determinant.

A universal construction for bialgebras associated with vector spaces and algebraic maps.

The Dubois-Violette and Launer Hopf algebra and co-quasi triangular properties are key to the proof.

Abstract

We prove that, in case $A(c)$ = the FRT construction of a braided vector space $(V,c)$ admits a weakly Frobenius algebra $\mathfrak B$ (e.g. if the braiding is rigid and its Nichols algebra is finite dimensional), then the Hopf envelope of $A(c)$ is simply the localization of $A(c)$ by a single element called the quantum determinant associated to the weakly Frobenius algebra. This generalizes a result of the author together with Gast\'on A. Garc\'ia in \cite{FG}, where the same statement was proved, but with extra hypotheses that we now know were unnecessary. On the way, we describe a universal way of constructing a universal bialgebra attached to a finite dimensional vector space together with some algebraic structure given by a family of maps $\{f_i:V^{\otimes n_i}\to V^{\otimes m_i}\}$. The Dubois-Violette and Launer Hopf algebra and the co-quasi triangular property of the FRT construction play a fundamental role on the proof.