Lifting of Nichols algebras of type \(A_2\) and pointed Hopf algebras of order \(p^4\) (Q2762031)

The authors present some results on the classification of pointed Hopf algebras with commutative coradical over an algebraically closed field \(k\) of characteristic \(0\). The method used is the ``lifting method'' which can be described briefly as follows. Let \(\Gamma\) be a finite Abelian group, and \(V\) a module in the category of \(k[\Gamma]\) Yetter-Drinfel'd modules. Suppose that \({\mathcal B}(V)\), the Nichols algebra of \(V\), is finite-dimensional. Then the Radford biproduct \(H={\mathcal B}(V)\#k[\Gamma]\) is a finite-dimensional pointed coradically graded Hopf algebra. A pointed Hopf algebra \(A\) with coradical \(k[\Gamma]\) is a lifting of \(H\) if \(\text{gr}(A)\), the associated graded Hopf algebra, is isomorphic to \(H\). For example, for \(V\) a quantum linear space, liftings of \(H\) as above were computed by \textit{N. Andruskiewitsch} and \textit{H.-J. Schneider} [J. Algebra 209, No. 2, 658-691 (1998; Zbl 0919.16027)], and these results yielded the description of all pointed Hopf algebras of dimension \(p^3\), \(p\) a prime. (See also \textit{S. Caenepeel} and \textit{S. Dăscălescu} [J. Algebra 209, No. 2, 622-634 (1998; Zbl 0917.16016)].) For \(p=2\) the classification of pointed Hopf algebras of dimension 16 has been done by S. Caenepeel, S. Dăscălescu and Ş. Raianu, and dimension 32 by M. Graña.NEWLINENEWLINENEWLINEIn order to study Hopf algebras of dimension \(p^4\), one must study liftings of bosonizations \(H={\mathcal B}(V)\#k[\Gamma]\) where the \(p\)-dimensional \(k\)-space \(V\in{^{k[\Gamma]}_{k[\Gamma]}\mathcal{YD}}\) is no longer a quantum linear space but has type \(A_2\) (meaning that \(V\) has Dynkin diagram \(A_2\)). This is a tricky computation handled in Section 3 of the paper. The main theorem of the paper states that every pointed Hopf algebra of dimension \(p^4\) is a lifting of \({\mathcal B}(V)\#k[\Gamma]\) where \(V\) is a quantum linear space if \(\Gamma\) has order \(p^3\) or \(p^2\) and \(V\) is of type \(A_2\) if \(\Gamma\) has order \(p\). The case of a lifting of \({\mathcal B}(V)\#k[C_{p^2}]\) is the only one in which infinite families of nonisomorphic Hopf algebras occur, and here it has been shown by \textit{A. Masuoka} [Proc. Am. Math. Soc. 129, No. 11, 3185-3192 (2001; Zbl 0985.16026)] that the members of these families are quasi-isomorphic.NEWLINENEWLINEFor the entire collection see [Zbl 0958.00024].