We investigate the first cohomology space associated with the embedding of the Lie superalgebra K(2) of contact vector fields on the (1,2)-dimensional supercircle S 1|2 in the Lie superalgebra SΨDO(S 1|2 ) of superpseudodifferential operators with smooth coefficients. Following Ovsienko and Roger, we show that this space is ten-dimensional with only even cocycles and we give explicit expressions of the basis cocycles.

We consider the action of the Lie algebra of polynomial vector fields, vect(1), by the Lie derivative on the space of symbols S n δ = n j=0 F δ−j . We study deformations of this action. We exhibit explicit expressions of some 2-cocycles generating the second cohomology space H 2 diff (vect(1), D ν,µ ) where D ν,µ is the space of differential operators from F ν to F µ . Necessary second-order integrability conditions of any infinitesimal deformations of S n δ are given. We describe completely the formal deformations for some spaces S n δ and we give concrete examples of non trivial deformations.

We classify deformations of the standard embedding of the Lie superalgebra K(2) of contact vector fields on the (1, 2)-dimensional supercircle into the Lie superalgebra SΨD(S 1|2 ) of pseudodifferential operators on the supercircle S 1|2 . The proposed approach leads to the deformations of the central charge induced on K(2) by the canonical central extension of SΨD(S 1|2 ).

We classify nontrivial deformations of the standard embedding of the Lie superalgebra K(1) of contact vector fields on the (1,1)-dimensional supercircle into the Lie superalgebra of superpseudodifferential operators on the supercircle. This approach leads to the deformations of the central charge induced on K(1) by the canonical central extension of SΨDO.