Low-dimensional Cohomology Groups Research Articles

On Cohomology Groups of Four-Dimensional Nilpotent Associative Algebras

The study of cohomology groups is one of the most intensive and exciting researches that arises from algebraic topology. Particularly, the dimension of cohomology groups is a highly useful invariant which plays a rigorous role in the geometric classification of associative algebras. This work focuses on the applications of low dimensional cohomology groups. In this regards, the cohomology groups of degree zero and degree one of nilpotent associative algebras in dimension four are described in matrix form.

On the low-dimensional cohomology groups of the IA-automorphism group of the free group of rank three

AbstractIn this paper, we study the structure of the rational cohomology groups of the IA-automorphism group$\mathrm {IA}_3$of the free group of rank three by using combinatorial group theory and representation theory. In particular, we detect a nontrivial irreducible component in the second cohomology group of$\mathrm {IA}_3$, which is not contained in the image of the cup product map of the first cohomology groups. We also show that the triple cup product of the first cohomology groups is trivial. As a corollary, we obtain that the fourth term of the lower central series of$\mathrm {IA}_3$has finite index in that of the Andreadakis–Johnson filtration of$\mathrm {IA}_3$.

Abelian extensions and crossed modules of Hom-Lie algebras

In this paper we study the low dimensional cohomology groups of Hom-Lie algebras and their relation with derivations, abelian extensions and crossed modules. On one hand, we introduce the notion of α-abelian extensions and we obtain a five term exact sequence in cohomology. On the other hand, we study crossed modules of Hom-Lie algebras showing their equivalence with cat1-Hom-Lie algebras, and we introduce α-crossed modules to have a better understanding of the third cohomology group.

On Contractions of Three-Dimensional Complex Associative Algebras

Contraction is one of the most important concepts that motivated by numerous applications in different fields of physics and mathematics. In this work, the contractions of complex associative algebras are considered. We focus on the variety A3() of all complex associative algebras of dimension three (including nonunital). Various contractions criteria are collected and new criteria are proposed to test the possible existence of contraction for each pair of associative algebras. One of the main tools is the use of the low-dimensional cohomology groups of these algebras. As a result, we prove that the variety A3() has seven irreducible components, two of dimension 5, four of dimension 7 and one of dimension 9.

Structure of The Planar Galilean Conformal Algebra

In this paper, we compute the low-dimensional cohomology groups of the planar Galilean conformal algebra introduced by Bagchi and Goparkumar. Consequently we determine its derivations, central extensions, and automorphisms.

On the cohomology of Leibniz conformal algebras

We construct a new cohomology complex of Leibniz conformal algebras with coefficients in a representation instead of a module. The low-dimensional cohomology groups of this complex are computed. Meanwhile, we construct a Leibniz algebra from a Leibniz conformal algebra and prove that the category of Leibniz conformal algebras is equivalent to the category of equivalence classes of formal distribution Leibniz algebras.

Low-dimensional cohomology of Lie superalgebra [formula omitted] with coefficients in Witt or Special superalgebras

Over a field of characteristic p>2, the Witt superalgebras are viewed as modules of the special linear Lie superalgebra slm|n by means of the adjoint representation. The low-dimensional cohomology groups of slm|n with coefficients in Witt superalgebras are computed by virtue of the direct sum decomposition of submodules and the weight space decompositions of these submodules relative to the standard Cartan subalgebra of slm|n. Then we are able to obtain the low-dimensional cohomology groups of slm|n with coefficients in Special superalgebras, which are slm|n-submodules of Witt superalgebras.

LOW-DIMENSIONAL COHOMOLOGY OF LIE SUPERALGEBRA $A(1, 0)$ WITH COEFFICIENTS IN WITT OR SPECIAL SUPERALGEBRAS

Over a field of characteristic $p \gt 2$; the low-dimensional cohomology groups of the special linear Lie superalgebra $\mathbf{A}(1;0)$ with coefficients in Witt superalgebra are computed by means of a direct sum decomposition of submodules and the weight space decomposition of the Witt superalgebra viewed as $\mathbf{A}(1;0)$-module. As application, the lowdimensional cohomology groups of $\mathbf{A}(1;0)$ with coefficients in Special superalgebra are also determined.

A seven-term exact sequence for the cohomology of a group extension

In this paper we construct a seven-term exact sequence involving the cohomology groups of a group extension. Although the existence of such a sequence can be derived using spectral sequence arguments, there is little knowledge about some of the maps occurring in the sequence, limiting its usefulness. Here we present a construction using only very elementary tools, always related to the notion of conjugation in a group. This results in a complete and usable description of all the maps, which we describe both on cocycle level as on the level of the interpretations of low dimensional cohomology groups (e.g. group extensions).

The prolongation of central extensions

The aim of this paper is to study the $(alpha‎, ‎gamma)$-prolongation of central extensions‎. ‎We obtain the obstruction theory for $(alpha‎, ‎gamma)$-prolongations and classify $(alpha‎, ‎gamma)$-prolongations thanks to low-dimensional cohomology groups of groups‎.

COMPUTATION OF LOW-DIMENSIONAL (CO)HOMOLOGY GROUPS FOR INFINITE SEQUENCES OF p-GROUPS WITH FIXED COCLASS

Eick and Leedham-Green introduced infinite sequences of p-groups of fixed coclass. Here we describe an algorithm to compute the Schur multiplicators of all groups in an infinite sequence simultaneously. Based on this, we prove that these Schur multiplicators can be described by a single parametrised presentation; this confirms a conjecture by Eick. Similar results are obtained for certain low-dimensional cohomology groups; these results support a conjecture by Carlson.

Low-Dimensional Cohomology Groups of the Lie Algebras W(a, b)

In this article, we determine one-dimensional (Leibniz) central extensions, the derivation algebra, and the automorphism group of the infinite-dimensional Lie algebra W(a, b) (for a, b ∈ ℂ) which is the semidirect product Lie algebra of the Witt algebra and its tensor density module I(a, b).

Deformations of W1( n)⊗ A and modular semisimple Lie algebras with a solvable maximal subalgebra

In one of his last papers, Boris Weisfeiler proved that if modular semisimple Lie algebra possesses a solvable maximal subalgebra which defines in it a long filtration, then associated graded algebra is isomorphic to one constructed from the Zassenhaus algebra tensored with the divided powers algebra. We completely determine such class of algebras, calculating in process low-dimensional cohomology groups of Zassenhaus algebra tensored with any associative commutative algebra.

Obstructions to clifford system extensions of algebras

In this paper we do phrase the obstruction for realization of a generalized group character, and then we give a classification of Clifford systems in terms of suitable low-dimensional cohomology groups.

On cohomology of Lie algebras

for all o1, * ,an in L. This definition makes Cn(L, M) into a G-module and further ,y (8f)=6(y yf) where a is the coboundary operator: Cn(L, M) -*Cn+'(L, M) so that the cohomology groups Hn(L, M) take on the structure of a G-module. Professor Hochschild suggested to the author that he define an operation of G on the standard interpretations of the low dimensional cohomology groups which would agree with the operation of G on Hn(L, M) as given above. This problem has an analogue in the cohomology of groups. There, if L is a normal subgroup of a group G and f is any function of Ln+' to M, the operation