The structure of Hopf algebras with a weak projection (Q1841202)

The author studies the structure of Hopf algebras that have a weak projection. Let \(H'\) be a Hopf subalgebra of the bialgebra \(H\) over a field \(k\). A map \(\pi\colon H\to H'\) is a weak (left) projection if \(\pi\) is a left \(H'\)-linear coalgebra map and \(\pi|_{H'}=\text{id}_{H'}\). In this situation, \(Q=H^{\prime+}H\) is a right \(H'\)-comodule coalgebra with the \(H'\)-comodule structure given by \(\rho'(\overline h)=\overline h_{[0]}\otimes\overline h_{[1]}=\overline h_{(2)}\otimes S(\pi(h_{(1)}))\pi(h_{(2)})\), and \(\psi=(\pi\otimes\nu)\Delta\colon H\to H'\times Q\) is an \(H'\)-module coalgebra isomorphism, where \(H'\times Q\) is the cosmash product and \(\nu\colon H\to Q\) is the canonical epimorphism. Let \(H'\) be a bialgebra and \(Q\) a right \(H'\)-module coalgebra with the comodule structure given by \(Q\ni q\mapsto q_{[0]}\otimes q_{[1]}\in Q\otimes H'\). Then one can form a cosmash product \(H'\times Q\). Assume there are maps \(Q\otimes H'\ni q\otimes x\mapsto q\leftharpoonup x\in Q\), \(Q\otimes H'\ni q\otimes x\mapsto q\rightharpoonup x\in H'\), \(Q\otimes Q\ni p\otimes q\mapsto p\cdot q\in Q\) and \(Q\otimes Q\ni p\otimes q\mapsto\tau(p|q)\in H'\), and an element \(1\in Q\). The paper describes equivalent conditions for \(H'\times Q\) to be a bialgebra with unit \(1\times 1\) and multiplication \[ (x\times p)(y\times q)=x(p_{(1)}\rightharpoonup y_{(1)})\tau(p_{(2)}\leftharpoonup y_{(2)}|q_{(1)[0]})\times (p_{(3)}\leftharpoonup y_{(3)}q_{(1)[1]})\cdot q_{(2)}, \] such that the map \(j\colon H'\ni x\mapsto x\times 1\in H'\times Q\) is a bialgebra map satisfying \(\pi j=\text{id}_{H'}\), where \(\pi\colon H'\times Q\ni x\times q\mapsto x\varepsilon(q)\in H'\). It is also proved that every bialgebra structure on \(H'\times Q\) such that \(H'\) is a subbialgebra of \(H'\times Q\) and \(\pi\colon H'\times Q\to H'\) is a left \(H'\)-linear coalgebra map with \(\pi|_{H'}=\text{id}_{H'}\) is of the form described above. Furthermore, let \(H'\times Q\) be a bialgebra as above. Then \(H'\times Q\) is a Hopf algebra if and only if \(H'\) is a Hopf algebra, and the map \(\beta_Q\colon Q\times Q\ni p\otimes q\mapsto p\cdot q_{(1)}\otimes q_{(2)}\in Q\otimes Q\) is a bijection.