Symmetries of modules of differential operators on the supercircle $$S^{1|n} $$ S 1 | n

J. Boujelben (), I. Safi (), Z. Saoudi () and K. Tounsi ()
Additional contact information
J. Boujelben: Faculté des sciences de Sfax
I. Safi: I.P.E.I.S
Z. Saoudi: Faculté des sciences de Sfax
K. Tounsi: Faculté des sciences de Sfax

Indian Journal of Pure and Applied Mathematics, 2022, vol. 53, issue 3, 701-719

Abstract: Abstract Let $$\mathfrak {F}_{\lambda }^n$$ F λ n be the space of tensor densities of degree $$\lambda \in \mathbb {C}$$ λ ∈ C on the supercircle $$S^{1|n}$$ S 1 | n . We consider the space $$\mathfrak {D}_{\lambda ,\mu }^{n,k}$$ D λ , μ n , k of k-th order linear differential operators from $$\mathfrak {F}_{\lambda }^n$$ F λ n to $$\mathfrak {F}_{\mu }^n$$ F μ n as a module over the superalgebra $$\mathcal {K}(n)$$ K ( n ) of contact vector fields on $$S^{1|n}$$ S 1 | n and we compute the superalgebra of endomrphisms on $$\mathfrak {D}_{\lambda ,\mu }^{n,k}$$ D λ , μ n , k commuting with the $$\mathfrak {aff}(n|1)$$ aff ( n | 1 ) -action where $$\mathfrak {aff}(n|1)$$ aff ( n | 1 ) is the affine subalgebra of $$\mathcal {K}(n)$$ K ( n ) . This result allows us to determine the superalgebra of endomrphisms on $$\mathfrak {D}_{\lambda ,\mu }^{n,k}$$ D λ , μ n , k commuting with the $$\mathfrak {osp}(n|2)$$ osp ( n | 2 ) -action for $$n\in \{1,2,3\}$$ n ∈ { 1 , 2 , 3 } where $$\mathfrak {osp}(n|2)$$ osp ( n | 2 ) is the orthosymlectic superalgebras of $$S^{1|n}$$ S 1 | n .

Keywords: Contact structure; Differential operators; Densities; 53D55 (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s13226-021-00164-y

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