On Bimeasurings

We introduce and study bimeasurings from pairs of bialge-bras to algebras. It is shown that the universal bimeasuring bialgebra construction, which arises from Sweedler's universal measuring coalgebra construction and generalizes the finite dual, gives rise to a contravariant functor on the category of bialgebras adjoint to itself. An interpretation of bimeasurings as algebras in the category of Hopf modules is considered. Measurings have first been introduced and studied my M.E. Sweedler [Sw]. They correspond to homomorphisms of algebras over a coalgebra which are cofree as comodules [GP]. There is a universal measuring coalgebra M (B, A) and measuring θ : M (B, A) ⊗ B − → A for every pair of algebras A and B such that C-measurings from B to A correspond bijectively to coalgebra maps from C to M (B, A). If B is a Hopf algebra and A is commutative then M (B, A) carries a natural Hopf algebra structure [Ma]. If in addition, C is a Hopf algebra then one may consider maps ψ : C ⊗ B − → A which measure in both variables C and B. In the cocommutative case these bimeasurings account for the " mixed " term in the second Sweedler cohomology group as shown in [Ma]. If A is commutative then universal bimeasuring Hopf algebras (and universal bimeasuring) B(C, A) and B(B, A) exist so that bimeasurings θ : C ⊗ B − → A bijectively correspond to Hopf algebra maps from C to B(B, A) as well as Hopf algebra maps from B to B(C, A). In fact and hence the functor B(, A) on the category of Hopf algebras is adjoint to itself. In the special case A = k this gives a new proof that the finite dual construction • = B(, k) is adjoint to itself [Ta] (see [Mi] for the proof). Moreover, there is a natural injective map which is always an isomorphism in the cocommutative case, and restricts to the well known isomorphism C • ⊗ B • ≃ (C ⊗ B) • when A = k.