Question

The question is about commutator in integral forms. Let $A$ an associative algebra over a field of characteristic zero, $x\in A$ and $k\in \mathbb Z$, we denote $x^{(k)}=\frac{x^{k}}{k!}$. How to calculate brackets $[x_\beta^{(n)}, x_\alpha^{(m)}]$? Can I proceed doing the calculation for $[x_\beta^{n}, x_\alpha^{m}]$ and then multiply by $\frac{1}{n!m!}$ and try to rewrite the result in a divided power notation?

ADDED: Let $g$ be a finite-dimensional simple Lie algebra and ${\alpha_i,...,\alpha_n}$ the set of positive roots. It is known that there exist an integral form for $U(g)$ and it is generated by $(x_{\alpha}^-)^{(k)}$ where $k>0$ and $\alpha$ is a positive root.

Answer 1

As I understand, your question is about relations on the generators of the Garland integral form for the hyper loop algebra of $\mathfrak g$. Maybe the following papers will help you:

H.Garland, The arithmetic theory of loop algebras, J. Algebra 53 (1978), 480--551.

D. Jakelic, A. Moura, Finite-dimensional representations of hyper loop algebras, Pacific J. Math. 233 (2007), 371--402.