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More on crossed modules of Lie, Leibniz, associative and diassociative algebras  [PDF]
José Manuel Casas,Rafael F. Casado,Emzar Khmaladze,Manuel Ladra
Mathematics , 2015,
Abstract: Adjoint functors between the categories of crossed modules of 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.
Lie crossed modules and gauge-invariant actions for 2-BF theories  [PDF]
Joao Faria Martins,Aleksandar Mikovic
Physics , 2010,
Abstract: We generalize the BF theory action to the case of a general Lie crossed module $(H \to G)$, where $G$ and $H$ are non-abelian Lie groups. Our construction requires the existence of $G$-invariant non-degenerate bilinear forms on the Lie algebras of $G$ and $H$ and we show that there are many examples of such Lie crossed modules by using the construction of crossed modules provided by short chain complexes of vector spaces. We also generalize this construction to an arbitrary chain complex of vector spaces, of finite type. We construct two gauge-invariant actions for 2-flat and fake-flat 2-connections with auxiliary fields. The first action is of the same type as the BFCG action introduced by Girelli, Pfeiffer and Popescu for a special class of Lie crossed modules, where $H$ is abelian. The second action is an extended BFCG action which contains an additional auxiliary field. However, these two actions are related by a field redefinition. We also construct a three-parameter deformation of the extended BFCG action, which we believe to be relevant for the construction of non-trivial invariants of knotted surfaces embedded in the four-sphere.
Pointed Homotopy of Maps Between 2-Crossed Modules of Commutative Algebras  [PDF]
?. ?lker Ak?a,Kadir Emir,Jo?o Faria Martins
Mathematics , 2014,
Abstract: We address the homotopy theory of 2-crossed modules of commutative algebras, which are equivalent to simplicial commutative algebras with Moore complex of length two. In particular, we construct for maps of 2-crossed modules a homotopy relation, and prove that it yields an equivalence relation in very unrestricted cases (freeness up to order one of the domain 2-crossed module). This latter condition strictly includes the case when the domain is cofibrant. Furthermore, we prove that this notion of homotopy yields a groupoid with objects being the 2-crossed module maps between two fixed 2-crossed modules (with free up to order one domain), the morphisms being the homotopies between 2-crossed module maps.
On Hopf 2-algebras  [PDF]
Yael Fregier,Friedrich Wagemann
Mathematics , 2009,
Abstract: Our main goal in this paper is to translate the diagram relating groups, Lie algebras and Hopf algebras to the corresponding 2-objects, i.e. to categorify it. This is done interpreting 2-objects as crossed modules and showing the compatibility of the standard functors linking groups, Lie algebras and Hopf algebras with the concept of a crossed module. One outcome is the construction of an enveloping algebra of the string Lie algebra of Baez-Crans, another is the clarification of the passage from crossed modules of Hopf algebras to Hopf 2-algebras.
Hom-Lie 2-algebras  [PDF]
Yunhe Sheng,Danhua Chen
Mathematics , 2011, DOI: 10.1016/j.jalgebra.2012.11.032
Abstract: In this paper, we introduce the notions of hom-Lie 2-algebras, which is the categorification of hom-Lie algebras, $HL_\infty$-algebras, which is the hom-analogue of $L_\infty$-algebras, and crossed modules of hom-Lie algebras. We prove that the category of hom-Lie 2-algebras and the category of 2-term $HL_\infty$-algebras are equivalent. We give a detailed study on skeletal hom-Lie 2-algebras. In particular, we construct the hom-analogues of the string Lie 2-algebras associated to any semisimple involutive hom-Lie algebras. We also proved that there is a one-to-one correspondence between strict hom-Lie 2-algebras and crossed modules of hom-Lie algebras. We give the construction of strict hom-Lie 2-algebras from hom-left-symmetric algebras and symplectic hom-Lie algebras.
Categorification of Pre-Lie Algebras and Solutions of 2-graded Classical Yang-Baxter Equations  [PDF]
Yunhe Sheng
Mathematics , 2014,
Abstract: In this paper, we introduce the notion of a pre-Lie 2-algebra, which is a categorification of a pre-Lie algebra. We prove that the category of pre-Lie 2-algebras and the category of 2-term pre-Lie$_\infty$-algebras are equivalent. We classify skeletal pre-Lie 2-algebras by the third cohomology of a pre-Lie algebra. We prove that crossed modules of pre-Lie algebras are in one-to-one correspondence with strict pre-Lie 2-algebras. $\mathcal O$-operators on Lie 2-algebras are introduced, which can be used to construct pre-Lie 2-algebras. As an application, we give solutions of 2-graded classical Yang-Baxter equations in some semidirect product Lie 2-algebras.
Computing 2-Dimensional Algebras: Crossed Modules and Cat1-Algebras  [PDF]
Z. Arvasi,A. Odabas
Mathematics , 2013,
Abstract: In this paper, we described the GAP implementation of crossed modules of commutative algebras and cat$^{1}$-algebras and their equivalence.
On 3-crossed modules of algebras  [PDF]
T. S. Kuzp? nar?,A. Odaba?,E. ?. Uslu
Mathematics , 2010,
Abstract: 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 dimensional homological information and show the existence of this resolution for an arbitrary $\mathbf{k}$-algebra.
Integrating morphisms of Lie 2-algebras  [PDF]
Behrang Noohi
Mathematics , 2009, DOI: 10.1112/S0010437X1200067X
Abstract: We show how to integrate a weak morphism of Lie algebra crossed-modules to a weak morphism of Lie 2-groups. To do so we develop a theory of butterflies for 2-term L_infty algebras. In particular, we obtain a new description of the bicategory of 2-term L_infty algebras. We use butterflies to give a functorial construction of connected covers of Lie 2-groups. We also discuss the notion of homotopy fiber of a morphism of 2-term L_infty algebras.
Crossed Modules of a Class of Five-Dimensional Lie Algebras

Pure Mathematics (PM) , 2019, DOI: 10.12677/PM.2019.93035
The set of equivalence classes of crossed modules of four-dimensional unsolvable Lie algebras and the third relative cohomology groups were studied in [1]. Based on this work, the present paper will study the crossed modules of a certain five-dimensional Lie algebra and determine the condition of its equivalent class. Furthermore, it is shown that the third relative cohomology group of the five-dimensional Lie algebra is not trivial.
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