A theory of highest weight modules over an arbitrary nite-dimensional Lie su-peralgebra is constructed. A necessary and suucient condition for the nite-dimensionality of such modules is proved. Generic nite-dimensional irreducible representations are deened and an explicit character formula for such representations is written down. It is conjectured that this formula applies to any generic nite...

Drinfeld twist is applied to the Lie algebra gl(2) so that a two-parametric deformation of it is obtained, which is identical to the Jordanian deformation of the gl(2) obtained by Aneva et al. The same twist element is applied to deform the Lie superalgebra sl(1/2), since the gl(2) is embedded into the sl(1/2). By making use of the FRT-formalism, we construct a deformation of the Lie supergroup...

We propose to parametrize the configuration space of one-dimensional quantum systems of N identical particles by the elementary symmetric polynomials of bosonic and fermionic coordinates. It is shown that in this parametrization the Hamiltonians of the AN , BCN , BN , CN and DN Calogero and Sutherland models, as well as their supersymmetric generalizations, can be expressed — for arbitrary valu...

We show that a primeness criterion for enveloping algebras of Le superalgebras discovered by Bell is applicable to the Cartan type Lie superal-gebras W(n), n even. Other algebras are considered but there are no deenitive answers in these cases. Allen Bell has shown in B] that if L is a nite-dimensional Lie superalgebra over a eld of characteristic zero, then the primeness of the universal envel...

We present explicit free field representations for the N = 4 doubly extended superconformal algebra, Ãγ. This algebra generalizes and contains all previous N = 4 superconformal algebras. We have found Ãγ to be obtained by hamiltonian reduction of the Lie superalgebra D(2|1;α). In addition, screening operators are explicitly given and the associated singular vectors identified. We use this to pr...

We already showed that Betti numbers are all zero when w is not equal to h for (w,h)-doubly weighted homology groups of some special pre Lie superalgebra and the first number 0 = h. In this note, we show second

It is shown that the direct limit of semistandard decomposition tableau model for polynomial representations queer Lie superalgebra exists, which believed to be crystal upper half corresponding quantum group. An extension this describe combinatorially given. Furthermore, it may recovered from in most cases.

Abstract We prove that an integral block of the category $\mathcal O$ periplectic Lie superalgebra contains a non-semisimple Jantzen middle if and only it simple module atypical highest weight. As consequence, every does not admit Kazhdan–Lusztig theory in sense Cline, Parshall, Scott.

Following Feigin and Fuchs, we compute the first cohomology of Lie superalgebra $\mathcal{K}(1)$ contact vector fields on (1,1)-dimensional real superspace with coefficients in linear differential operators acting superspaces weighted densities. We also same, but $\mathfrak{osp}(1|2)$-relative, cohomology. explicitly give 1-cocycles spanning these classify generic formal $\mathfrak{osp}(1|2)$-t...