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.

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...

Integrability of the one-dimensional Hubbard model and factorised scattering problem encountered on worldsheet AdS strings can be expressed in terms a peculiar quantum algebra. In this article, we derive classical limit these algebraic integrable structures based established results for exceptional simple Lie superalgebra d(2,1;epsilon) along with standard sl(2) which form supersymmetric isomet...

We construct certain modified highest weight modules which are called quasi in this paper. Using the modules, we introduce a new category of over an affine Lie superalgebra contains projective covers. also prove that both these covers and satisfy vanishing property BRST cohomology.

A relation between $\frac{1}{2}$-derivations of Lie algebras and transposed Poisson was established. Some non-trivial with a certain algebra (Witt algebra, $\mathcal{W}(a,-1)$, thin solvable abelian nilpotent radical) were constructed. In particular, we constructed an example the associative parts isomorphic to Laurent polynomials Witt algebra. On other side, it proven that there are no part se...