Related papers: Crossed modules for Lie 2-algebras
In this paper we aim to understand the category of stable-Yetter-Drinfeld modules over enveloping algebra of Lie algebras. To do so, we need to define such modules over Lie algebras. These two categories are shown to be isomorphic. A mixed…
Over a field of characteristic p>2, firstly, the structure of Kac modules of Lie superalgebra $\tilde{P}(2)$ and the weight space decompositions are given. Secondly, the weight-derivations of $\tilde{P}(2)$ to its Kac modules are computed.…
An algebraic category $\mathcal{C}$ is called balanced if the cotriple cohomology of any object of $\mathcal{C}$ vanishes in positive dimensions on injective coefficient modules. Important examples of balanced and of non-balanced categories…
The extending structures problem for Zinbiel 2-algebras is studied. We introduce the concept of unified products for Zinbiel 2-algebras. Some special cases of unified products such as crossed product and matched pair of Zinbiel 2-algebras…
A well-known and old result of Hazewinkel and Koszul states that the cohomology of a finite-dimensional Lie algebra is isomorphic, up to a suitable shift, to its twisted homology, a Lie-theoretical version of Poincare duality. This paper…
We introduce for any Poisson algebra a bicomplex of free Poisson modules, and use it to show that the Poisson cohomology theory introduced in the paper "[M. Flato, M. Gerstenhaber and A. A. Voronov, Cohomology and Deformation of Leibniz…
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…
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…
The aim here is to sketch the development of ideas related to brackets and similar concepts: Some purely group theoretical combinatorics due to Ph. Hall led to a proof of the Jacobi identity for the Whitehead product in homotopy theory.…
We firstly prove the completeness of the category of crossed modules in a modified category of interest. Afterwards, we define pullback crossed modules and pullback cat$^1$-objects that are both obtained by pullback diagrams with extra…
In this dissertation, we investigate the cohomology theory of restricted Lie algebras. The representation theory of restricted Lie algebras is reviewed including a description of the restricted universal enveloping algebra. In the case of…
A general procedure is presented which associates to a finite crossed module a premodular category, generalizing the representation categories of a finite group and of its double, and the extent to which the resulting category fails to be…
In this paper we present the Sweedler cohomology for a cocommutative weak Hopf algebra H. We show that the second cohomology group classifies completely the weak crossed products, having a common preunit, of H with a commutative left…
In this paper, we describe a regular representation given by Cayley theorem for 2-crossed modules of groups and their associated Gray 3-group groupoids with a single 0-cell and equivalently cat2-groups.
We give a full classification of Lie algebras of specific type in complexified Clifford algebras. These sixteen Lie algebras are direct sums of subspaces of quaternion types. We obtain isomorphisms between these Lie algebras and classical…
In the paper we introduce the notion of twisted derivation of a bialgebra. Twisted derivations appear as infinitesimal symmetries of the category of representations. More precisely they are infinitesimal versions of twisted automorphisms of…
We define two $(n+1)$ graded Lie brackets on spaces of multilinear mappings. The first one is able to recognize $n$-graded associative algebras and their modules and gives immediately the correct differential for Hochschild cohomology. The…
A general theory of the Frolicher-Nijenhuis and Schouten-Nijenhuis brackets in the category of modules over a commutative algebra is described. Some related structures and (co)homology invariants are discussed, as well as applications to…
We introduce the notion of a crossed product of an algebra by a coalgebra $C$, which generalises the notion of a crossed product by a bialgebra well-studied in the theory of Hopf algebras. The result of such a crossed product is an algebra…
The aim of this paper is to construct triassociative algebras (from operators), new actions and crossed modules from a given one, and to make the connexion between these notions on Leibniz algebras or triassociative algebras and the…