Related papers: The third cohomology group classifies crossed modu…

We prove that the group of normalized cohomological invariants of degree 3 modulo the subgroup of semidecomposable invariants of a semisimple split linear algebraic group G is isomorphic to the torsion part of the Chow group of codimension…

We show that the class of connected, simple Lie groups that have non-vanishing third-degree continuous cohomology with trivial $\mathbb{R}$-coefficients consists precisely of all simple complex Lie groups and of…

Let A a k-algebra, H a Hopf algebra, E = A#H a general crossed product and M an E-bimodule. We obtain a complex simpler than the canonical one, giving the Hochschild homology of E with coefficients in M. This complex is eqquiped with a…

This paper is devoted to the study of the group ${\rm Ext} (G,H)$ of all extensions of topological abelian groups $0\to H\to X\to G\to 0$ and the group ${\rm Ext}_{TVS} (Z,Y)$ of all extensions of topological vector spaces $0\to Y\to X\to…

Given a spectral triple $(A,H,D)$ and a $C^*$-dynamical system $(\mathbf{A}, G, \alpha)$ where $A$ is dense in $\mathbf{A}$ and $G$ is a locally compact group, we extend the triple to a triplet $(\mathcal{B},\mathcal{H},\mathcal{D})$ on the…

We compute the rational cohomology of the moduli space of non-singular complex projective curves of genus 3 with two marked points.

In this paper, we consider compatible Hom-Lie triple systems. Compatible Hom-Lie triple systems are characterized as Maurer-Cartan elements in a suitable bidifferential graded Lie algebra. We also define a cohomology theory for compatible…

In this paper the third homology group of the linear group GL_2(R) with integral coefficients is investigated, where R is a commutative ring with many units.

Given a field $\mathbb K$, for any $n\geq 3$ the first cohomology group $H^1(G_n,A^*_n)$ of the special linear group $G_n = \mathrm{SL}(n,{\mathbb K})$ over the dual $A^*_n$ of its adjoint module $A_n$ is isomorphic to the space…

The codomain category of a generalized homology theory is the category of modules over a ring. For an abelian category A, an A-valued (generalized) homology theory is defined by formally replacing the category of modules with the category…

In this paper, we calculate cohomology of a classical Lie algebra of type $A_2$ over an algebraically field $k$ of characteristic $p=3$ with coefficients in simple modules. To describe their structure, we will consider them as modules over…

In this paper we define 3-crossed modules for commutative (Lie) algebras and investigate the relation between this construction and the simplicial algebras. Also we define the projective 3-crossed resolution for investigate a higher…

We compute the group cohomology of $32\Gamma_3f$, a certain group of order 32. For this we construct explicit cocycle representatives of the cohomology generators. We thus lay to rest a discrepancy between several published computations of…

The category of modules over the endomorphism algebra of a rigid object in a Hom-finite triangulated category C has been given two different descriptions: On the one hand, as shown by Osamu Iyama and Yuji Yoshino, it is equivalent to an…

We compute cohomology of the moduli space of genus three curves with level two structure and some related spaces. In particular, we determine the cohomology groups of the moduli space of plane quartics with level two structure as…

Let $A$ be a $(G, \chi)$-Hopf algebra with bijection antipode and let $M$ be a $G$-graded $A$-bimodule. We prove that there exists an isomorphism \mathrm{HH}^*_{\rm gr}(A, M)\cong{\rm Ext}^*_{A{-}{\rm gr}} (\K, {^{ad}(M)}), where $\K$ is…

Let X, Y, and Z be topological modules over a topological ring R. In this paper, we introduce three different classes of bounded bigroup homomorphisms from X \times Y into Z with respect to the three different uniform convergence…

In this paper we study the cohomology of (strict) Lie 2-groups. We obtain an explicit Bott-Shulman type map in the case of a Lie 2-group corresponding to the crossed module $A\to 1$. The cohomology of the Lie 2-groups corresponding to the…

We show that for any set of primes $\mathcal{P}$ there exists a space $M_{\mathcal{P}}$ which is a homology and cohomology 3-manifold with coefficients in $\mathbb{Z}_{p}$ for $p\in \mathcal{ P}$ and is not a homology or cohomology…

In this paper we state some applications of Gr-category theory on the classification of crossed modules and on the classification of extensions of groups of the type of a crossed module.