##### Sesión Álgebra No Conmuntativa y Homológica

Diciembre 14, 11:20 ~ 11:40

## Nichols algebras that are quantum planes

### JURY GIRALDI, João Matheus

Recently, in \cite[Prop. 4.8, 4.9]{GGi}, braided vector spaces $(V, c)$ of dimension $2$ and non-diagonal type were found but such that the Nichols algebras are quantum planes. Consequently, the following question arises naturally: classify all the Nichols algebras (of rigid braided vector spaces) that are isomorphic to quantum linear spaces as algebras. In this paper, we solve this question for quantum planes. More specifically, the classification of the solutions of the quantum Yang--Baxter equation had already been performed by J. Hietarinta when $\dim V = 2$ \cite{Hi}. Thus, we consider the associated braided vector spaces and compute the quadratic relations. Therefore, we classify all these Nichols algebras that have at least one quadratic relation. This is a joint work with N. Andruskiewitsch \cite{AGi}. \addtocontents{toc}{\protect\setcounter{tocdepth}{0}} \begin{thebibliography}{99} \bibitem[AGi]{AGi} N. Andruskiewitsch and J. M. J. Giraldi. \emph{ Nichols algebras that are quantum planes}, arXiv: 1702.02506. To appear in Linear Multilinear Algebra. \bibitem[GGi]{GGi} G. A. García and J. M. J. Giraldi, \emph{On Hopf Algebras over quantum subgroups}, arXiv: 1605.03995. \bibitem[Hi]{Hi} J. Hietarinta. \emph{Solving the two-dimensional constant quantum Yang-Baxter equation}, J. Math. Phys. \textbf{34}, (1993). \end{thebibliography}

Autores: JURY GIRALDI, João Matheus / ANDRUSKIEWITSCH, Nicolás.