Small bialgebras with a projection. (Q2456198)

Let \(A\) be a bialgebra over a field \(K\) of characteristic zero, \(H\) a subbialgebra with antipode, such that there is a coalgebra projection \(p\) of \(A\) onto \(H\). Let \(R\) be the \(H\)-coinvariant part of \(A\), i.e., the elements \(a\) in \(A\) so that \(\sum a_1\otimes p(a_2)=a\otimes 1\). The authors determine the structure of \(A\) when \(R\) is connected with one-dimensional space of primitive elements, and \(A\) is either finite-dimensional or cosemisimple. The precise results are too technical to present here, but we will mention some of the ingredients of the description. \(R\) is a pre-bialgebra in the braided category of left \(H\)-Yetter-Drinfeld modules, i.e., a coalgebra in this category with a multiplication and unit satisfying certain properties differing from a bialgebra in the category in that, in general, the multiplication need not be associative nor a morphism of \(H\)-comodules. However under the authors' assumptions, the multiplication on \(R\) will be associative. Another key idea is that of a Yetter-Drinfeld datum for \(q\), a primitive \(N\)-th root of unity. It is a triple \((H,g,X)\), \(g\) a group-like element of \(H\), \(X\) a character on \(H\), \(X(g)=q\), and the sums \(gX(h_1)h_2\) and \(h_1X(h_2)g\) are equal for all \(h\) in \(H\). In this case, the graded algebra \(K[x]/(X^N)\) is a braided Hopf algebra in the category of left \(H\)-Yetter-Drinfeld modules, called a quantum line. If \(R\) has a one-dimensional space \(Ky\) of primitive elements, then \(R\) has a divided power sequence \(d_0=1,d_1=y,d_2,\dots,d_{N-1}\) related to the Yetter-Drinfeld datum. One section of the paper discusses pre-bialgebras with a cocycle. In the main theorems, all this is used to describe a specific basis of \(A\), which will itself be a Hopf algebra. This study is related to the idea of biproducts and bosonization, and to the lifting method in the work of \textit{N. Andruskiewitsch} and \textit{H.-J. Schneider} [J. Algebra 209, No. 2, 658--691 (1998; Zbl 0919.16027)].