Hopf modules and their duals (Q5932502)

Let \(A\) be a bialgebra over a field \(k\). Every finite dimensional \(k\)-vector space \(V\) induces a free right \(A\)-module \(V_A=V\otimes_k A\). Every representation \(\lambda\colon A\to\text{End}_k(V)\) gives an \(A\)-bimodule structure on \(V_A\), where the left \(A\)-module structure is defined by \(a\cdot(v\otimes b)\lambda(a_{(1)})v\otimes a_{(2)}b\), where \(\Delta(a)=a_{(1)}\otimes a_{(2)}\), which makes \(V_A\) a right covariant bimodule in the sense of \textit{S. L. Woronowicz} [Commun. Math. Phys. 122, No. 1, 125-170 (1989; Zbl 0751.58042)]. The Main Theorem 5.3 asserts that the dual anti-representation \(\widetilde\lambda\colon A\to\text{End}_k(\widetilde V)\), where \(\widetilde V\) is the dual vector space of \(V\), generates a right \(A\)-covariant bimodule structure on \(_A\widetilde V\). The authors deduce a consequence (Corollary 5.4) on the dual of a bicovariant bimodule generated by a Yetter-Drinfeld module over a bialgebra with a bijective antipode. The authors say that Corollary 5.4 can be relevant for adapting the vector field formalism [\textit{A. Borowiec}, Czech. J. Phys. 46, No. 12, 1197-1202 (1996; Zbl 1020.58501), ibid. 47, No. 11, 1093-1100 (1997; Zbl 0948.46051)] to the case of Woronowicz's differential calculus on quantum groups.