Motivation. Few years ago I constructed a family of internal Hopf algebras in the Loday-Pirashvili tensor category of linear maps which is in a sense a generalization of the algebra of regular functions on the group $GL(n)$. A student of mine (Matija Bašić) has worked out certain internal analogue of a Weyl algebra in that context and one could conceivably make much of the story on algebraic groups and invariant differential operators in that setup. However, we did not continue for a while and this project stalled in a way. Its motivation was to find a geometric theory of integration of Leibniz algebras in characteristic zero, and more generally Lie algebras in Loday-Pirashvili tensor category. There is an Ado theorem in this setup, so it is sufficient to do integration for Lie subalgebras in a Loday-Pirashvili analogue of $gl(n)$ (here $n$ generalizes to data of a linear map between two finite dimensional vector spaces). For the latter there is an internal version (as well as another version) of a universal enveloping Hopf algebra. On the other hand, there is my analogue of the function Hopf algebra on the $GL$ in Loday-Pirashvili category. I have a picture of certain modification of scheme theory to make it into an algebraic group in certain abstract context, but so far this is not fully done. Thus I would like to prove, not from abstract principles but by direct check that the two Hopf algebras are dual one to another.
But trouble: in classical case of the usual Lie groups and enveloping algebras I am aware only of the proofs using what a Lie group is and what its Lie algebra is, while my GL is given as a Hopf algebra somewhat alike usual functions on $GL(n)$, with just a little more exotic relations. So what I should generalize is down to earth explicit proof that $U(gl_n)$ and $\mathcal{O}(GL(n))$ are dual as Hopf algebras. Emphasis is on Hopf. Notice that it is not sufficient to write down the pairing for multiplicative generators, but for the whole vector space basis: the algebras are not free and to predict the pairing between higher order monomials does not follow from knowing pairing just between matrix elements $t^i_j$ and the generators of the Lie algebras $gl_n\subset U(gl_n)$. But it should not be that difficult.
Question: Do you know how to do algebraically prove that we have a Hopf pairing (possibly with some sort of formulas for the pairing) between classical $GL(n)$ and $U(gl_n)$ ? We should never use any knowledge on $GL(n)$ and $gl_n$ except their generators and relations, as most of other facts are nontrivial to generalize to my problem which motivates the question.
Related questions: explicit-isomorphism-between-distributions-and-universal-enveloping-algebra and hopf-algebra-structure-on-the-universal-enveloping-algebra-of-a-leibniz-algebra.
