Deforming the orthosymplectic Lie superalgebra inside the Lie superalgebra of superpseudodifferential operators

Abstract We classify deformations of the standard embedding of the Lie algebra s l ( 2 ) into both the Lie algebra Ψ D O L of pseudodifferential operators with polynomial coefficients and the Poisson Lie algebra P , we prove that any formal deformation is equivalent to its infinitesimal part. We study also the super analogue of this problem for the case of the standard embedding of the orthosymplectic Lie superalgebra o s p ( n | 2 ) on the ( 1 , n ) -dimensional superspace R 1 | n into the Lie superalgebra S Ψ D O ( n ) of superpseudodifferential operators with polynomial coefficients, where n = 1 , 2 getting the necessary and sufficient conditions for its integrability. Finally, by using the contract procedure we deduce similar results for the standard embedding into the Poisson Lie superalgebra S P ( n ) .