Crossed squares and 2-crossed modules

预交叉模与rack同调

Pre-crossed modules and rack homology

本文为预交叉模定义同调理论，并在其由rack自由生成时特化为rack同调，从而统一了相关同调概念。

论李代数交叉模的中心

On the centre of crossed modules of Lie algebras

该文探究李代数交叉模和其中心的关联，证明任意李代数交叉模都能对应到包含两个同伦李代数上同调的正合序列，为李代数交叉模的中心研究提供了同调视角的结论。

论 Gottlieb 群、Drinfeld 中心与交叉模的中心

On the Gottlieb group, Drinfeld centre and the centre of a crossed module

本文引入交叉模的中心概念，利用交叉同态建立其与群上同调、Gottlieb 群、Drinfeld 中心及 Whitehead 中心的联系。

A comonadic interpretation of Baues-Ellis homology of crossed modules

We introduce and study a homology theory of crossed modules with coefficients in an abelian crossed module. We discuss the basic properties of these new homology groups and give some applications. We then restrict our attention to the case of integral coefficients. In this case we regain the homology of crossed modules originally defined by Baues and further developed by Ellis. We show that it is an instance of Barr-Beck comonadic homology, so that we may use a result of Everaert and Gran to obtain Hopf formulae in all dimensions.

On crossed product rings with twisted involutions, their module categories and L-theory

We study the Farrell-Jones Conjecture with coefficients in an additive G-category with involution. This is a variant of the L-theoretic Farrell-Jones Conjecture which originally deals with group rings with the standard involution. We show that this formulation of the conjecture can be applied to crossed product rings R*G equipped with twisted involutions and automatically implies the a priori more general fibered version.

On Lie algebra crossed modules

This article constructs a crossed module corresponding to the generator of the third cohomology group with trivial coefficients of a complex simple Lie algebra. This generator reads as <[,],>, constructed from the Lie bracket [,] and the Killing form <,>. The construction is inspired by the corresponding construction for the Lie algebra of formal vector fields in one formal variable on R, and its subalgebra sl_2(R), where the generator is usually called Godbillon-Vey class.

非交换 $H^2$ 与在群中的系数及在交叉模中的系数

Nonabelian $H^2$ with coefficients in a group and with coefficients in a crossed module

本文扩展了非交换 $H^1$ 的上同调精确序列，通过在交叉模中使用非交换 $H^2$ 来研究非交换上同调。

$q$-交叉模与李代数的 $q$-可容性

$q$-crossed modules and $q$-capability of Lie algebras

本文利用李代数的非阿贝尔 $q$-外积，提出并研究两种李代数 $q$-可容性概念，分别关联 $q$-交叉模及 Ellis 的群论版本。

群在交叉模系数下的非阿贝尔上同调与伽罗瓦上同调

Non-abelian hypercohomology of a group with coefficients in a crossed module, and Galois cohomology

发展了群在交叉模系数下的上同调理论，并应用于定义约化代数群伽罗瓦上同调的阿贝尔化映射。

Cyclic Homology of Hopf Comodule Algebras and Hopf Module Coalgebras

In this paper we construct a cylindrical module $A \natural \mathcal{H}$ for an $\mathcal{H}$-comodule algebra $A$, where the antipode of the Hopf algebra $\mathcal{H}$ is bijective. We show that the cyclic module associated to the diagonal of $A \natural \mathcal{H}$ is isomorphic with the cyclic module of the crossed product algebra $A \rtimes \mathcal{H}$. This enables us to derive a spectral sequence for the cyclic homology of the crossed product algebra. We also construct a cocylindrical module for Hopf module coalgebras and establish a similar spectral sequence to compute the cyclic coho…

Hopf Algebra Equivariant Cyclic Homology and Cyclic Homology of Crossed Product Algebras

We introduce the cylindrical module $A \natural \mathcal{H}$, where $\mathcal{H}$ is a Hopf algebra and $A$ is a Hopf module algebra over $\mathcal{H}$. We show that there exists an isomorphism between $\mathsf{C}_{\bullet}(A^{op} \rtimes \mathcal{H}^{cop})$ the cyclic module of the crossed product algebra $A^{op} \rtimes \mathcal{H}^{cop} $, and $Δ(A \natural \mathcal{H}) $, the cyclic module related to the diagonal of $A \natural \mathcal{H}$. If $S$, the antipode of $\mathcal{H}$, is invertible it follows that $\mathsf{C}_{\bullet}(A \rtimes \mathcal{H}) \simeq Δ(A^{op} \natural \mathcal{H}…

Cyclic Cohomology of Crossed Coproduct Coalgebras

We extend our work in~\cite{rm01} to the case of Hopf comodule coalgebras. We introduce the cocylindrical module $C \natural^{} \mathcal{H}$, where $\mathcal{H}$ is a Hopf algebra with bijective antipode and $C$ is a Hopf comodule coalgebra over $\mathcal{H}$. We show that there exists an isomorphism between the cocyclic module of the crossed coproduct coalgebra $C > \blacktriangleleft \mathcal{H} $ and $Δ(C \natural^{}\mathcal{H}) $, the cocyclic module related to the diagonal of $C \natural^{} \mathcal{H}$. We approximate $HC^{\bullet}(C > \blacktriangleleft \mathcal{H}) $ by a spectral sequ…

Crossed modules, non-abelian extensions of associative conformal algebras and Wells exact sequences

In this paper, we introduce the notions of crossed module of associative conformal algebras, 2-term strongly homotopy associative conformal algebras, and discuss the relationship between them and the 3-th Hochschild cohomology of associative conformal algebras. We classify the non-abelian extensions by introducing the non-abelian cohomology. We show that non-abelian extensions of an associative conformal algebra can be viewed as Maurer-Cartan elements of a suitable differential graded Lie algebra, and prove that the Deligne groupoid of this differential graded Lie algebra corresponds one to on…

Cyclic homology of Hopf Galois extensions and Hopf algebras

Let H be a Hopf algebra. By definition a modular crossed H-module is a vector space M on which H acts and coacts in a compatible way. To every modular crossed H-module M we associate a cyclic object Z(H,M). The cyclic homology of Z(H,M) extends the usual cyclic homology of the algebra structure of H, and the relative cyclic homology of an H-Galois extension. For a Hopf subalgebra K we compute, under some assumptions, the cyclic homology of an induced modular crossed module. As a direct application of this computation, we describe the relative cyclic homology of strongly graded algebras. In par…