Bitensorial Hopf algebra

For any field k of zero characteristic we give a functor from the category of k-vector spaces into the category of k-Hopf algebras, attaching to any vector space V its bitensorial pointed Hopf algebra A(V). This Hopf algebra is graded, fulfills a universal property, and contains a remarkable subspace P of primitive elements, which as a conjecture may generate the Lie algebra Prim A(V). In case V is finite-dimensional we exhibit a Hopf pairing between A(V) and A(V) whose kernel contains the (Hopf) ideal generated by the elements of P of degree greater than or equal to 2.