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...

We describe the structure of a Verma module with a generic highest weight at the critical level over a symmetrizable affine Lie superalgebra ĝ 6= A(2k, 2l). We obtain the character formula for a simple module with a generic highest weight at the critical level conjectured by V. G. Kac and D. A. Kazhdan.

We study the hamiltonian reduction of affine Lie superalgebra sl(2|1)(1). Based on a scalar Lax operator formalism, we derive the free field realization of the classical topological topological algebra which appears in the c ≤ 1 non-critical strings. In the quantum case, we analyze the BRST cohomology to get the quantum free field expression of the algebra.

In this paper, we study the simple modules for the restricted Lie superalgebra gl(m|n). A condition for the simplicity of the induced modules is given, and an analogue of Kac-Weisfeiler theorem is proved. Mathematics Subject Classification (2000): 17B50; 17B10.

The famous Drinfeld-Kohno theorem for simple Lie algebras states that the monodromy representation of the Knizhnik-Zamolodchikov equations for these Lie algebras expresses explicitly via R-matrices of the corresponding Drinfeld-Jimbo quantum groups. This result was generalized by the second author to simple Lie superalgebras of type A-G. In this paper, we generalize the Drinfeld-Kohno theorem t...