Starting from a Hopf algebra endowed with an action of a group π by Hopf automorphisms, we construct (by a “twisted” double method) a quasitriangular Hopf π-coalgebra. This method allows us to obtain non-trivial examples of quasitriangular Hopf π-coalgebras for any finite group π and for infinite groups π such as GLn(k). In particular, we define the graded quantum groups, which are Hopf π-coalg...

We prove the quasi-Hopf algebra version of the Nichols-Zoeller theorem: A finite-dimensional quasi-Hopf algebra is free over any quasi-Hopf subalgebra.

Let $H$ be a Hopf algebra and $A$ an $H$-bimodule algebra‎. ‎In this paper‎, ‎we investigate Gorenstein global dimensions for Hopf‎ ‎algebras and twisted smash product algebras $Astar H$‎. ‎Results from‎ ‎the literature are generalized‎. 

In this work, a differential delay equation (DDE) with a cubic nonlinearity is analyzed as two parameters are varied by means of a center manifold reduction. This reduction is applied directly to the case where the system undergoes a Hopf-Hopf bifurcation. This procedure replaces the original DDE with four first-order ODEs, an approximation valid in the neighborhood of the Hopf-Hopf bifurcation...

Hopf representation is a module and comodule with a consistency condition that is more general than the consistency condition of Hopf modules. For a Hopf algebra H, we construct an induced Hopf representation from a representation of a bialgebra B using a bialgebra epimorphism π : H → B. Application on the quantum group Eq(2) is given. Mathematics Subject Classification: 16T05, 06B15, 17B37

In this paper, we consider controlling and anti-controlling Hopf bifurcations in discrete maps using feedback controller of polynomial functions. It is shown that such a polynomial feedback controller is easy to be implemented, which not only preserves the system s equilibrium solutions, but also keeps the dimension of the system unchanged. Examples are used to show, with this type of controlle...

For G a finite abelian group, we study the properties of general equivalence relations on Gn = Gn Sn , the wreath product of G with the symmetric group Sn , also known as the G-coloured symmetric group. We show that under certain conditions, some equivalence relations give rise to subalgebras of kGn as well as graded connected Hopf subalgebras of ⊕ n≥o kGn . In particular we construct a G-colou...

We give a new construction of a Hopf subalgebra of the Hopf algebra of Free quasi-symmetric functions whose bases are indexed by objects belonging to the Baxter combinatorial family (i.e. Baxter permutations, pairs of twin binary trees, etc.). This construction relies on the definition of the Baxter monoid, analog of the plactic monoid and the sylvester monoid, and on a Robinson-Schensted-like ...

The goal of this paper is to give a new method of constructing finite-dimensional semisimple triangular Hopf algebras, including minimal ones which are non-trivial (i.e. not group algebras). The paper shows that such Hopf algebras are quite abundant. It also discovers an unexpected connection of such Hopf algebras with bijective 1-cocycles on finite groups and set-theoretical solutions of the q...