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

Mabrouk Ben Ammar () and Rabeb Sidaoui
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Mabrouk Ben Ammar: Université de Sfax
Rabeb Sidaoui: Université de Sfax

Indian Journal of Pure and Applied Mathematics, 2021, vol. 52, issue 4, 1263-1275

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

Keywords: Cohomology; Weighted Densities; 17B56 (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1007/s13226-021-00012-z

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