On Lie algebra crossed modules

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.

The third cohomology group classifies crossed module extensions

We give an elementary proof of the well-known fact that the third cohomology group H^3(G, M) of a group G with coefficients in an abelian G-module M is in bijection to the set Ext^2(G, M) of equivalence classes of crossed module extensions of G with M.

$q$-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。

Crossed modules and symmetric cohomology of groups

This paper links the third symmetric cohomology (introduced by Staic and Zarelua ) to crossed modules with certain properties. The equivalent result in the language of 2-groups states that an extension of 2-groups corresponds to an element of $HS^3$ iff it possesses a section which preserves inverses in the 2-categorical sense. This ties in with Staic's (and Zarelua's) result regarding $HS^2$ and abelian extensions of groups.

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。

交叉模与代数上同调的operad方法

Crossed modules and cohomology of algebras over an operad

本研究提出代数operad上P-代数的n-交叉模一般定义，验证其在经典operad下与已有定义的一致性，并建立其等价类阿贝尔群与高阶operadic上同调群的自然同构关系。

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…

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…

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。

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…