Cohomology of %${\mathfrak {sl}}(2)%$ acting on the space of n-ary differential operators on %${\mathbb {R}}%$

Abstract We consider the spaces %${\mathcal {F}}_\mu%$ of polynomial %$\mu%$-densities on the line as %${\mathfrak {sl}}(2)%$-modules and then we compute the cohomological spaces %$\text {H}^1_\text {diff}({\mathfrak {sl}}(2), {\mathcal {D}}_{{\bar{\lambda }},\mu })%$, where %$\mu \in {\mathbb {R}}%...
Ausführliche Beschreibung

Abstract We consider the spaces %${\mathcal {F}}_\mu%$ of polynomial %$\mu%$-densities on the line as %${\mathfrak {sl}}(2)%$-modules and then we compute the cohomological spaces %$\text {H}^1_\text {diff}({\mathfrak {sl}}(2), {\mathcal {D}}_{{\bar{\lambda }},\mu })%$, where %$\mu \in {\mathbb {R}}%$, %${\bar{\lambda }}=(\lambda _1,\dots ,\lambda _n) \in {\mathbb {R}}^n%$ and %${\mathcal {D}}_{{\bar{\lambda }},\mu }%$ is the space of n-ary differential operators from %${\mathcal {F}}_{\lambda _1}\otimes \cdots \otimes {\mathcal {F}}_{\lambda _n}%$ to %${\mathcal {F}}_\mu%$. Ausführliche Beschreibung