We provide an analog of Tannaka Theory for Hopf algebras in the context of crossed Hopf group coalgebras introduced by Turaev. Following Street and our previous work on the quantum double of crossed structures, we give a construction, via Tannaka Theory, of the quantum double of crossed Hopf group algebras (not necessarily of finite type).

An explicit general formula is proposed for controlling Hopf bifurcation using state feedback. This method can be used to either delay (or even eliminate) an existing Hopf bifurcation or change a subcritical Hopf bifurcation to supercritical. The Lorenz system is used to illustrate the application of the formula.

Let D(H) be the quantum double associated to a finite dimensional quasi-Hopf algebra H, as in [9] and [10]. In this note, we first generalize a result of Majid [15] for Hopf algebras, and then prove that the quantum double of a finite dimensional quasitriangular quasi-Hopf algebra is a biproduct in the sense of [4].

We give the classification of (co-)path Hopf algebras and semi-path Hopf algebras with pointed module structures. This leads to the classification of multiple crown algebras and multiple Taft algebras as well as kG-Yetter-Drinfeld modules and Nichols algebras with pointed module structures. Moreover, we characterize quantum enveloping algebras in terms of semi-path Hopf algebras. 2000 Mathemati...

The left and right homological integrals are introduced for a large class of infinite dimensional Hopf algebras. Using the homological integrals we prove a version of Maschke’s theorem for infinite dimensional Hopf algebras. The generalization of Maschke’s theorem and homological integrals are the keys to studying noetherian regular Hopf algebras of Gelfand-Kirillov dimension one.

The main result in this paper states that every strongly graded bialgebra whose component of grade 1 is a finite-dimensional Hopf algebra is itself a Hopf algebra. This fact is used to obtain a group cohomology classification of strongly graded Hopf algebras, with 1-component of finite dimension, from known results on strongly graded bialgebras.  2002 Elsevier Science (USA)

We give the classification of (co-)path Hopf algebras and semi-path Hopf algebras with pointed module structures. This leads to the classification of multiple crown algebras and multiple Taft algebras as well as pointed Yetter-Drinfeld kG-modules and the corresponding Nichols algebras. Moreover, we characterize quantum enveloping algebras in terms of semi-path Hopf algebras.

In this work we study the induction theory for Hopf group coal-gebra. To reach this goal we define a substructure B of a Hopf group coalgebra H, called subHopf group coalgebra. Also, we introduced the definition of Hopf group suboalgebra and group coisotropic quantum subgroup of H.

We introduce C∗-pseudo-multiplicative unitaries and (concrete) Hopf C∗-bimodules, which are C∗-algebraic variants of the pseudo-multiplicative unitaries on Hilbert spaces and the Hopf-von Neumann-bimodules studied by Enock, Lesieur, and Vallin [5, 6, 4, 10, 19, 20]. Moreover, we associate to every regular C∗-pseudo-multiplicative unitary two Hopf-C∗bimodules and discuss examples related to loca...

In this note the notion of kernel of a representation of a semisimple Hopf algebra is introduced. Similar properties to the kernel of a group representation are proved in some special cases. In particular, every normal Hopf subalgebra of a semisimple Hopf algebra H is the kernel of a representation of H