# Distribution algebras and loop algebras

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.

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Loop algebra is infinite-dimensional, so it is not clear for me how will you define the convolution of functions - you need Feynman integral - this will not come for free - you will have "anomaly" which is related to the central extension of the loop algebra and "critical level"... –  Alexander Chervov Jan 31 '12 at 7:30
@Alexander: I guess Chris is referring to the "hyperalgebra" treatment by Garland which is parallel to the treatment of "distribution algebras" in the finite dimensional case. –  Jim Humphreys Mar 7 '12 at 23:27

Not exaclty what you want, but the closest I can formally think is the following statement (which you can find correctly stated in Frenkel-Ben-Zvi's book and is due to Beilinson-Drinfeld)

"The affine chiral algebra consists of delta functions supported at the unit of the Beilinson-Drinfeld Grassmanian".

Just to say why this has some remote connection with what you want: you may want to think of the affine chiral algebra $V^k(g)$ associated to a finite dimensional $g$ as a quotient of the universal enveloping algebra of $\hat{g} = g((t))\oplus \mathbb{C}K$

$$V^k(g) := U( \hat{g}) \otimes_{ U(g[ [t]] \oplus \mathbb{C} K)} \mathbb{C}$$

The Beilinson-Drinfeld Grassmanian is to $V^k(g)$ what the Lie group $G$ is to $g$.

Finally note that delta functions supported at the identity of $G$ is the same as distributions supported at the identity of $G$ and is the same as $U(g)$.

With the forgiveness of the experts for trying to fit these thoughts in a few lines.

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