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 super-algebra 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.

We find the free field construction of basic $W$-current and screening currents for deformed $W$-superalgebra ${\cal W}_{q,t}\bigl(A(M,N)\bigr)$ associated with Lie superalgebra type $A(M,N)$. Using this construction, we introduce higher $W$-currents obtain a closed set quadratic relations among them. These are independent choice Dynkin-diagrams $A(M,N)$, though not. This allows us to define by...

Using Howe duality we compute explicitly Kostant-type homology groups for a wide class of representations of the infinite-dimensional Lie superalgebra b gl∞|∞ and its classical subalgebras at positive integral levels. We also obtain Kostant-type homology formulas for the Lie algebra b gl∞ at negative integral levels. We further construct resolutions in terms of generalized Verma modules for the...

Given any simple Lie superalgebra g, we investigate the structure of an arbitrary simple weight g-module. We introduce two invariants of simple weight modules: the shadow and the small Weyl group. Generalizing results of Fernando and Futorny we show that any simple module is obtained by parabolic induction from a cuspidal module of a Levi subsuper-algebra. Then we classify the cuspidal Levi sub...

In [Zh], R. B. Zhang found a way to link certain formal deformations of the Lie algebra o(2l + 1) and the Lie superalgebra osp(1, 2l). The aim of this article is to reformulate the Zhang transformation in the context of the quantum enveloping algebrasà la Drinfeld-Jimbo and to show how this construction can explain the main theorem of [GL2]: the annihilator of a Verma module over the Lie supera...

Soon after parastatistics has been introduced [1], it was discovered that it has a deep algebraic structure. It turned out that any n pairs of parafermion operators generate the simple Lie algebra so(2n + 1) [2, 3], and n pairs of paraboson creation and annihilation operators b 1 , . . . , bn generate a Lie superalgebra [4], isomorphic to one of the basic classical Lie superalgebras in the clas...