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 which is also a right C-comodule. We find the necessary and sufficient conditions for two coalgebra crossed products be equivalent. We show that the two-dimens...

We show that almost split sequences in the category of comodules over a coalgebra Γ with finite-dimensional right-hand term are direct limits of almost split sequences over finite dimensional subcoalgebras. In previous work we showed that such almost split sequences exist if the right hand term has a quasifinitely copresented linear dual. Conversely, taking limits of almost split sequences over...

The generalized quantum double lattice realization of 2d topological orders based on Hopf algebras is discussed in this work. Both left-module and right-module constructions are investigated. ribbon operators the classification excitations representations discussed. To generalize model to a surface with boundaries defects, we present systematic construction boundary Hamiltonian domain wall Hami...

The coalgebra approach to the construction of classical integrable systems from Poisson coalgebras is reviewed, and the essential role played by symplectic realizations in this framework is emphasized. Many examples of Hamiltonians with either undeformed or q-deformed coalgebra symmetry are given, and their Liouville superintegrability is discussed. Among them, (quasi-maximally) superintegrable...

Let A and B be two Hopf algebras and R ∈ Hom(B ⊗ A, A ⊗ B), the twisted tensor product Hopf algebra A#RB was introduced by S. Caenepeel et al in [3] and further studied in our recent work [6]. In this paper we give the necessary and sufficient conditions for A#RB to be a Hopf algebra with a projection. Furthermore, a braided Hopf algebra A is constructed by twisting the multiplication of A thro...

We consider a special category of Hopf algebras, depending on parameters Σ which possess properties similar to the category of representations of simple Lie group with highest weight λ. We connect quantum groups to minimal objects in this categories—they correspond to irreducible representations in the category of representations with highest weight λ. Moreover, we want to correspond quantum gr...

A differentially recursive sequence over a differential field is of elements satisfying homogeneous equation with non-constant coefficients (namely, Taylor expansions the field) in algebra Hurwitz series. The main aim this paper to explore space all sequences given non-zero differential. We show that these form two-sided vector admits, canonical way, structure Hopf algebroid subfield constant e...

Consider a coring with exact rational functor, and a finitely generated and projective right comodule. We construct a functor (coinduction functor) which is right adjoint to the hom-functor represented by this comodule. Using the coinduction functor, we establish a bijective map between the set of representative classes of torsion simple right comodules and the set of representative classes of ...

Fix a prime p. Given a finite group G, let H(G) denote its mod p cohomology. In the early 1990’s, Henn, Lannes, and Schwartz introduced two invariants d0(G) and d1(G) of H∗(G) viewed as a module over the mod p– Steenrod algebra. They showed that, in a precise sense, H∗(G) is respectively detected and determined by H(CG(V )) for d ≤ d0(G) and d ≤ d1(G), with V running through the elementary abel...

This thesis is concerned with the definition and the study of properties of homotopic Hopf-Galois extensions in the category Ch 0 k of chain complexes over a field k, equipped with its projective model structure. Given a differential graded k-Hopf algebra H of finite type, we define a homotopic H-Hopf-Galois extension to be a morphism ' : B ! A of augmented H-comodule dg-k-algebras, where B is ...