Towards a new cohomology theory for strict Lie 2-groups (original) (raw)

Homological aspects of Lie algebra crossed modules

manuscripta mathematica, 2010

Internal homology theory in the category of crossed modules of Lie algebras is constructed and investigated. Its relationship with the Chevalley-Eilenberg homology is established in terms of a long exact homology sequence.

Equivariant Crossed Modules and Cohomology of Groups with Operators

Bulletin of the Korean Mathematical Society, 2015

In this paper we study equivariant crossed modules in its link with strict graded categorical groups. The resulting Schreier theory for equivariant group extensions of the type of an equivariant crossed module generalizes both the theory of group extensions of the type of a crossed module and the one of equivariant group extensions.

Categorical Results in the Theory of Two-Crossed Modules of Commutative Algebras

arXiv (Cornell University), 2011

In this paper we explore some categorical results of 2-crossed module of commutative algebras extending work of Porter in [18]. We also show that the forgetful functor from the category of 2-crossed modules to the category of k-algebras, taking {L, M, P, ∂2, ∂1} to the base algebra P , is fibred and cofibred considering the pullback (coinduced) and induced 2-crossed modules constructions, respectively. Also we consider free 2crossed modules as an application of induced 2-crossed modules. Categorical Results in the Theory of Two-Crossed Modules of Commutative Algebras 2 We end with an application which leads to link free 2-crossed modules with induced 2-crossed modules. Conventions Throughout this paper k will be a fixed commutative ring and R will be a kalgebra with identity. All algebras will be commutative and actions will be left and the right actions in some references will be rewritten by using left actions.

Categorical structures of Lie–Rinehart crossed module

TURKISH JOURNAL OF MATHEMATICS, 2019

In this paper we give constructions of pullback, finite product, finite limit, coproduct, colimit, pushout, etc. in a special full subcategory XMod/L of the category of Lie-Rinehart crossed modules.

More on crossed modules in Lie, Leibniz, associative and diassociative algebras

Journal of Algebra and Its Applications, 2016

Adjoint functors between the categories of crossed modules in dialgebras and Leibniz algebras are constructed. The well known relations between the categories of Lie, Leibniz, associative algebras and dialgebras are extended to the respective categories of crossed modules.

Abelian extensions and crossed modules of Hom-Lie algebras

Journal of Pure and Applied Algebra

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 introduce crossed modules of Hom-Lie algebras showing their equivalence with cat 1-Hom-Lie algebras, and we introduce α-crossed modules to have a better understanding of the third cohomology group.

Universal enveloping crossed module of a Lie crossed module

Homology, Homotopy and Applications, 2014

We construct a pair of adjoint functors between the categories of crossed modules of Lie and associative algebras, which extends the classical one between the categories of Lie and associative algebras. This result is used to establish an equivalence of categories of modules over a Lie crossed module and its universal enveloping crossed module.