Ideals for character Hopf algebras. (Q392483)

A character Hopf algebra is a Hopf algebra which is generated as an algebra by elements \(a_1,\ldots,a_\theta\) and an Abelian group \(G(A)=\Gamma\) of all group-like elements such that for all \(1\leq i\leq\theta\), there are \(g_i\in\Gamma\) and characters \(\chi_i\in\widehat\Gamma\) with NEWLINE\[NEWLINE\Delta(a_i)= a_i\otimes 1+g_i\otimes a_i\quad\text{ and }\quad ga_i=\chi_i(g)a_ig\qquad (\forall g\in\Gamma).NEWLINE\]NEWLINE They are all pointed. This class of character Hopf algebras is very important for the classification of pointed Hopf algebras and can be seen as generalized quantum groups.NEWLINENEWLINE In the paper under the review, the author gives a structural description of character Hopf algebras in terms of generators and relations and obtains an analogous statement for Nichols algebras of diagonal type. For this aim, the author shows that any character Hopf algebra is a quotient of the smash product of a free algebra and a group algebra. The author then applies the theory of Lyndon words and a result of Kharchenko, and the generators of the ideal to build up a Gröbner basis.