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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 $x^{(k)}=\frac{x^{k}}{k!}$. If I need to calculate some bracket like $[x_\beta^{(n)}, x_\alpha^{(m)}]$, how should I proceed? Can I proceed doing the calculation to $[x_\beta^{n}, x_\alpha^{m}]$ and then to 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, $\{\alpha_i,...,\alpha_n\}$ the set of positive roots. It is known that there exist an integral form for $U(g)$ and it is generated by $(x_{\alpha,r}^-)^{(k)}$ where $k>0$, $r\in Z$ and $\alpha$ is a positive root. How to calculate $[(x_{\alpha,r}^-)^{(k)}, (x_{\beta,s}^-)^{(m)}]$?

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.

Commutator in hyperalgebras....

If I need to calculate some bracket like $[x_\beta^{(n)}, x_\alpha^{(m)}]$, how should I proceed? Can I proceed doing the calculation to $[x_\beta^{n}, x_\alpha^{m}]$ , then I multiply by $\frac{1}{n!m!}$ and try to write the result in a divided power notation???

Thanks.

I meant: Let $g$ be a finite-dimensional simple Lie algebra, $\{\alpha_i,...,\alpha_n\}$ the set of positive roots. Take $U_F(g\otimes C[t,t^{-1}])$ the hyperalgebra associated to $g\otimes C[t,t^{-1}]$ over some field of positive characteristic. It is well-known that there exist an integral form for $U_F(g)$ and it is generated by $(x_{\alpha,r}^-)^{(k)}$ where $k>0$, $r\in Z$ and $\alpha$ is a positive root. So, how to calculate $[ (x_{\alpha,r}^-)^{(k)}, (x_{\beta,s}^-)^{(m)}]$ ???? Is it clear now?

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 $x^{(k)}=\frac{x^{k}}{k!}$. If I need to calculate some bracket like $[x_\beta^{(n)}, x_\alpha^{(m)}]$, how should I proceed? Can I proceed doing the calculation to $[x_\beta^{n}, x_\alpha^{m}]$ and then to 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, $\{\alpha_i,...,\alpha_n\}$ the set of positive roots. It is known that there exist an integral form for $U(g)$ and it is generated by $(x_{\alpha,r}^-)^{(k)}$ where $k>0$, $r\in Z$ and $\alpha$ is a positive root. How to calculate $[(x_{\alpha,r}^-)^{(k)}, (x_{\beta,s}^-)^{(m)}]$?

Commutator in hyperalgebras.

Given an associative algera $A$ over a field os characteristic zero, $x\in A$ and $n\in \mathbb Z$, we define $x^{(k)}=\frac{x^k}{k!}$. If I need to calculate a bracket involving these divided powers, how should I proceed?

Thanks.

Commutator in hyperalgebras....

If I need to calculate some bracket like $[x_\beta^{(n)}, x_\alpha^{(m)}]$, how should I proceed? Can I proceed doing the calculation to $[x_\beta^{n}, x_\alpha^{m}]$ , then I multiply by $\frac{1}{n!m!}$ and try to write the result in a divided power notation???

Thanks.

I meant: Let $g$ be a finite-dimensional simple Lie algebra, $\{\alpha_i,...,\alpha_n\}$ the set of positive roots. Take $U_F(g\otimes C[t,t^{-1}])$ the hyperalgebra associated to $g\otimes C[t,t^{-1}]$ over some field of positive characteristic. It is well-known that there exist an integral form for $U_F(g)$ and it is generated by $(x_{\alpha,r}^-)^{(k)}$ where $k>0$, $r\in Z$ and $\alpha$ is a positive root. So, how to calculate $[ (x_{\alpha,r}^-)^{(k)}, (x_{\beta,s}^-)^{(m)}]$ ???? Is it clear now?