In our recent paper [DSK11] we computed the dimension of the variational Poisson cohomology H•K(V) for any quasiconstant coefficient l × l matrix differential operator K of order N with invertible leading coefficient, provided that V is a normal algebra of differential functions over a linearly closed differential field. In the present paper we show that, for K skewadjoint, the Z-graded Lie sup...

Abstract All results concern characteristic 2. We describe two procedures; each of which to every simple Lie algebra assigns a superalgebra. prove that finite-dimensional superalgebra is obtained as the result one these procedures. For algebras, in addition known “classical” restrictedness, we introduce (2,4)-structure on non-alternating series: orthogonal and Hamiltonian vector fields. superal...

We consider the supercircle S equipped with the standard contact structure. The conformal Lie superalgebra K(1) acts on S as the Lie superalgebra of contact vector fields; it contains the Möbius superalgebra osp(1|2). We study the space of linear differential operators on weighted densities as a module over osp(1|2). We introduce the canonical isomorphism between this space and the correspondin...

Let [Formula: see text] be a finite-dimensional simple Lie superalgebra of type text], or over text]. the simply connected semisimple algebraic group such that Suppose is nilpotent. We describe centralizer in and its center especially. also determine labeled Dynkin diagram for prove theorems relating dimension diagram.

Many work was done for filiform Lie algebras defined by M. Vergne [8]. An interesting fact is that this algebras are obtained by deformations of the filiform Lie algebra Ln,m. This was used for classifications in [4]. Like filiform Lie algebras, filiform Lie superalgebras are obtained by nilpotent deformations of the Lie superalgebra Ln,m. In this paper, we recall this fact and we study even co...

We introduce the notion of a Lie-like algebra⋄ (superalgebra⋄) for ⋄ ∈ { , , 3−rd }. By bundling a family of algebras, we introduce six new generalizations of Lie algebras which are called Lie-like algebras or Lie-like superalgebras in this paper. The examples of the new generalizations of Lie algebras come from an invariant algebra (or an invariant superalgebra), where the notion of an invaria...

The universal central extensions and their extension kernels of the matrix Lie superalgebra sl(m, n,A), the Steinberg Lie superalgebra st(m, n,A) in category SLeib of Leibniz superalgebras are determined under a weak assumption (compared with [MP]) using the first Hochschild homology and the first cyclic homology group.

Let Γ be a countable abelian semigroup and A be a countable abelian group satisfying a certain finiteness condition. Suppose that a group G acts on a (Γ × A)-graded Lie superalgebra L = ⊕ (α,a)∈Γ×A L(α,a) by Lie superalgebra automorphisms preserving the (Γ × A)-gradation. In this paper, we show that the Euler-Poincaré principle yields the generalized denominator identity for L and derive a clos...

We propose a new realization, using Harish-Chandra bimodules, of the Serre functor for the BGG category O associated to a semi-simple complex finite dimensional Lie algebra. We further show that our realization carries over to classical Lie superalgebras in many cases. Along the way we prove that category O and its parabolic generalizations for classical Lie superalgebras are categories with fu...