Low-dimensional Cohomology Research Articles

Schur multiplier and Schur covers of relative Rota–Baxter groups

Relative Rota–Baxter groups are generalizations of Rota–Baxter groups and share a close connection with skew left braces. These structures are well-known for offering bijective non-degenerate set-theoretical solutions to the Yang–Baxter equation. This paper builds upon the recently introduced extension theory and low-dimensional cohomology of relative Rota–Baxter groups. We prove an analogue of the Hochschild–Serre exact sequence for central extensions of relative Rota–Baxter groups. We introduce the Schur multiplier MRRB(A) of a relative Rota–Baxter group A=(A,B,β,T), and prove that the exponent of MRRB(A) divides |A||B| when both A and B are finite. We define weak isoclinism of relative Rota–Baxter groups, introduce their Schur covers, and prove that any two Schur covers of a finite bijective relative Rota–Baxter group are weakly isoclinic. The results align with recent results of Letourmy and Vendramin for skew left braces.

On low dimensional cohomology of crossed modules with nontrivial coefficients

On low dimensional cohomology of crossed modules with nontrivial coefficients

Second cohomology group and quadratic extensions of metric Hom-Jacobi–Jordan algebras

In this paper, we introduce and study the low dimensional cohomology of metric Hom-Jacobi–Jordan algebras. We establish one-to-one correspondence between the equivalence classes of abelian quadraticextensions of a Hom-Jacobi–Jordan algebra and its second cohomology group.

Restricted Cohomology of Restricted Lie Superalgebras

Suppose the ground field \(\mathbb{F}\) is an algebraically closed field characteristic of p > 2. In this paper, we investigate the restricted cohomology theory of restricted Lie superalgebras. Algebraic interpretations of low dimensional restricted cohomology of restricted Lie superalgebra are given. We show that there is a family of restricted model filiform Lie superalgebra \(L_{p,p}^\lambda\) structures parameterized by elements \(\lambda \in {\mathbb{F}^p}\). We explicitly describe both the 1-dimensional ordinary and restricted cohomology superspaces of \(L_{p,p}^\lambda\) with coefficients in the 1-dimensional trivial module and show that these superspaces are equal. We also describe the 2-dimensional ordinary and restricted cohomology superspaces of \(L_{p,p}^\lambda\) with coefficients in the 1-dimensional trivial module and show that these superspaces are unequal.

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.

Cohomology and deformations of Filippov algebroids

In this article, we study the deformations of Filippov algebroids. First, we define a differential graded Lie algebra for a Filippov algebroid by introducing the notion of Filippov multiderivations for a vector bundle. We then discuss deformations of a Filippov algebroid in terms of low-dimensional cohomology associated with this differential graded Lie algebra. We define Nijenhuis operators on Filippov algebroids and characterize trivial deformations of Filippov algebroids in terms of these operators. Finally, we define finite order deformations and discuss the problem of extending a given finite order deformation to a deformation of a higher order.

Vandermonde Varieties, Mirrored Spaces, and the Cohomology of Symmetric Semi-algebraic Sets

Let $\mathrm{R}$ be a real closed field. We prove that for each fixed $\ell, d \geq 0$, there exists an algorithm that takes as input a quantifier-free first order formula $\Phi$ with atoms $P=0, P > 0, P < 0 \text{ with } P \in \mathcal{P} \subset \mathrm{D}[X_1,\ldots,X_k]^{\mathfrak{S}_k}_{\leq d}$, where $\mathrm{D}$ is an ordered domain contained in $\mathrm{R}$, and computes the ranks of the first $(\ell+1)$ cohomology groups, of the symmetric semi-algebraic set defined by $\Phi$. The complexity of this algorithm (measured by the number of arithmetic operations in $\mathrm{D}$) is bounded by a \emph{polynomial} in $k$ and $\mathrm{card}(\mathcal{P})$ (for fixed $d$ and $\ell$). This result contrasts with the $\mathbf{PSPACE}$-hardness of the problem of computing just the zero-th Betti number (i.e. the number of semi-algebraically connected components) in the general case for $d \geq 2$ (taking the ordered domain $\mathrm{D}$ to be equal to $\mathbb{Z}$). The above algorithmic result is built on new representation theoretic results on the cohomology of symmetric semi-algebraic sets. We prove that the Specht modules corresponding to partitions having long lengths cannot occur with positive multiplicity in the isotypic decompositions of low dimensional cohomology modules of closed semi-algebraic sets defined by symmetric polynomials having small degrees. This result generalizes prior results obtained by the authors giving restrictions on such partitions in terms of their ranks, and is the key technical tool in the design of the algorithm mentioned in the previous paragraph.

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$\mathrm {IA}_3$.

Derivations and Deformations of δ-Jordan Lie Supertriple Systems

Let T be a δ-Jordan Lie supertriple system. We first introduce the notions of generalized derivations and representations of T and present some properties. Also, we study the low-dimensional cohomology and the coboundary operator of T, and then we investigate the deformations and Nijenhuis operators of T by choosing some suitable cohomologies.

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.

Hom-Lie-Rinehart algebras

ABSTRACTWe introduce hom-Lie-Rinehart algebras as an algebraic analogue of hom-Lie algebroids, and systematically describe a cohomology complex by considering coefficient modules. We define the notion of extensions for hom-Lie-Rinehart algebras. In the sequel, we deduce a characterization of low dimensional cohomology spaces in terms of the group of automorphisms of certain abelian extension and the equivalence classes of those abelian extensions in the category of hom-Lie-Rinehart algebras, respectively. We also construct a canonical example of hom-Lie-Rinehart algebra associated to a given Poisson algebra and an automorphism.

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.

An Explicit Seven-Term Exact Sequence for the Cohomology of a Lie Algebra Extension

We construct a seven-term exact sequence involving low degree cohomology spaces of a Lie algebra 𝔤, an ideal 𝔥 of 𝔤, and the quotient 𝔤/𝔥 with coefficients in a 𝔤-module. The existence of such a sequence follows from the Hochschild–Serre spectral sequence associated to the Lie algebra extension. However, some of the maps occurring in this induced sequence are not always explicitly known or easy to describe. In this article, we give alternative maps that yield an exact sequence of the same form, making use of the interpretations of the low-dimensional cohomology spaces in terms of derivations, extensions, etc. The maps are constructed using elementary methods. This alternative approach to the seven term exact sequence can certainly be useful, especially since we include straightforward cocycle descriptions of the constructed maps.

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 Hom-Lie algebras and q-deformed W(2, 2) algebra

This paper aims to study low dimensional cohomology of Hom-Lie algebras and the q-deformed W(2, 2) algebra. We show that the q-deformed W(2, 2) algebra is a Hom-Lie algebra. Also, we establish a one-to-one correspondence between the equivalence classes of one-dimensional central extensions of a Hom-Lie algebra and its second cohomology group, leading us to determine the second cohomology group of the q-deformed W(2, 2) algebra. In addition, we generalize some results of derivations of finitely generated Lie algebras with values in graded modules to Hom-Lie algebras. As application, we compute all αk-derivations and in particular the first cohomology group of the q-deformed W(2, 2) algebra.

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.