The algebra of distribution and its relationship with the universal enveloping algebra is discussed in the Jantzen's book, as we can see a discussion in the question link (more specifically, the Jim Humphreys comments).

The universal enveloping algebra and the hyperalgebra (i.e the distribution algebra) of an algebraic group over an algebraically closed field of characteristic zere coincides, however in positive characteristic it is not true.

When we are in the context of loop algebras and their subalgebras of fixed points under a finite order automorphism, is there a relationship between the universal enveloping algebra and something like a distribution algebra?

My motivation is because for these two cases we do have integral forms analogues to that of Kostant.