Cohomology Of Lie Superalgebra Research Articles

The Multiplier and Cohomology of Lie Superalgebras

Suppose that [Formula: see text] is a Lie superalgebra over a field [Formula: see text] of characteristic different from 2 and 3. In this paper, the so-called 5-sequence of cohomology for a central extension of [Formula: see text] is constructed and is proved to be exact. Moreover, the multiplier of [Formula: see text] is proved to be isomorphic to the second cohomology group with coefficients in the trivial module of [Formula: see text]. Finally, an upper bound of the superdimension of the second cohomology group is given in the situation when [Formula: see text] is nilpotent and finite-dimensional.

A Program for Computing the Cohomology of Lie Superalgebras of Vector Fields

The cohomology of Lie (super)algebras has many important applications in mathematics and physics. At present, since the required algebraic computations are very tedious, the cohomology is explicitly computed only in a few cases for different classes of Lie (super)algebras. That is why application of computer algebra methods is important for this problem. We describe an algorithm (and its C implementation) for computing the cohomology of Lie algebras and superalgebras. In elaborating the algorithm, we focused mainly on the cohomology with coefficients in trivial, adjoint, and coadjoint modules for Lie (super)algebras of the formal vector fields. These algebras have many applications to modern supersymmetric models of theoretical and mathematical physics. As an example, we consider the cohomology of the Poisson algebra Po(2) with coefficients in the trivial module and present 3- and 5-cocycles found by a computer. Bibliography: 6 titles.

Computation of Cohomologies of Lie Superalgebras: Algorithm and Implementation

In this work, an algorithm and its implementation in C for computing cohomologies of Lie algebras and superalgebras are presented. Using the program developed, we computed certain cohomologies of the Poisson and Hamiltonian vector fields and of their analogs based on the odd Poisson bracket (antibracket).

Cohomology of Lie superalgebras and their generalizations

The cohomology groups of Lie superalgebras and, more generally, of ε Lie algebras, are introduced and investigated. The main emphasis is on the case where the module of coefficients is nontrivial. Two general propositions are proved, which help to calculate the cohomology groups. Several examples are included to show the peculiarities of the super case. For L=sl(1|2), the cohomology groups H1(L,V) and H2(L,V), with V a finite-dimensional simple graded L-module, are determined, and the result is used to show that H2(L,U(L)) [with U(L) the enveloping algebra of L] is trivial. This implies that the superalgebra U(L) does not admit any nontrivial formal deformations (in the sense of Gerstenhaber). Garland’s theory of universal central extensions of Lie algebras is generalized to the case of ε Lie algebras.

Homology and cohomology of Lie superalgebra $\mathfrak{sl}(2,1)$ with coefficients in the spaces of finite-dimensional irreducible representations

Homology and cohomology of Lie superalgebra $\mathfrak{sl}(2,1)$ with coefficients in the spaces of finite-dimensional irreducible representations

On homology and cohomology of Lie superalgebras with coefficients in their finite-dimensional-representations

On homology and cohomology of Lie superalgebras with coefficients in their finite-dimensional-representations

Cohomologies of Lie superalgebras

Cohomologies of Lie superalgebras