In this paper, using the equivalence between category of crossed modules algebras and algebra-algebroids, we will explore notions ideality factors for these algebraic structures. We give structure a two sided ideal an algebra-algebroid notion quotient algebra-algebroid. By considering algebra-algebroid, show that module corresponding to is Conversely, by taking module, also module.

In this paper, we present a generalization of the concept bimultiplication algebra by defining crossed modules in associative algebras. Using structure, construct actor module algebras, and obtain annihilators as kernel morphism from to its actor. Moreover, link between square, two-dimensional module.

We provide an analog of the Joyal-Street center construction and of the Kassel-Turaev categorical quantum double in the context of the crossed categories introduced by Turaev. Then, we focus or attention to the case of categories of representation. In particular, we introduce the notion of a YetterDrinfeld module over a crossed group coalgebra H and we prove that both the category of Yetter-Dri...

In this note we consider crossed modules of groups (N → G, G→ Aut(N)), as a homotopy version of the inclusion N ⊂ G of a normal subgroup. Our main observation is a characterization of the underlying map N → G of a crossed module, in terms of a simplicial group structure on the associated bar construction. This approach allows for “natural” generalizations to other monoidal categories, in partic...

We exhibit crossed modules (i.e. certain 4 term exact sequences) corresponding to the Godbillon-Vey generator for W1, V ect(S ) and V ect1,0(Σ). Introduction There is a well known correspondence in homological algebra between (equivalence classes of) exact sequences Λ-modules, starting in M and ending in N with nmodules in between, and elements of ExtΛ(N,M). For Λ = U(g) the universal envelopin...

We introduce the cylindrical module A♮H, where H is a Hopf algebra with S2 = idH and A is a Hopf module algebra over H. We show that there exists a cyclic map between the cyclic module of the crossed product algebra A⋊H and ∆(A♮H), the cyclic module related to the diagonal of A♮H. In the cocommutative case, ∆(A♮H) ∼= C•(A ⋊H). Finally we approximate ∆(A♮H) by a spectral sequence and we give an ...

We construct canonical integral transforms, analogous to the Fourier transform, that have periods six and three. The existence of this transform is shown to arise naturally from the expectation that the Schwartz space on the real line, viewed as the Heisenberg module of Rieffel and Connes over the rotation C*-algebra, should extend to a module action over the crossed product of the latter by th...