• DOI:10.1007/s10468-022-10122-9
• Corpus ID: 221266358

Universal Quantum (Semi)groups and Hopf Envelopes

@article{Farinati2022UniversalQ,
  title={Universal Quantum (Semi)groups and Hopf Envelopes},
  author={Marco A. Farinati},
  journal={Algebras and Representation Theory},
  year={2022}
}

• M. Farinati
• Published 23 August 2020
• Mathematics
• Algebras and Representation Theory

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.