In this paper, we introduce the concept of a [Formula: see text]-Hopf algebra as Hopf obtained partial smash product and its base field, show that every is algebra. Moreover, method to compute actions given on field developed and, an application, exhibit all such type for some families algebras.

We show that every modular category is equivalent as an additive ribbon category to the category of finite-dimensional comodules of a Weak Hopf Algebra. This Weak Hopf Algebra is finite-dimensional, split cosemisimple, weakly cofactorizable, coribbon and has trivially intersecting base algebras. In order to arrive at this characterization of modular categories, we develop a generalization of Ta...

By weakening the counit and antipode axioms of a C∗-Hopf algebra and allowing for the coassociative coproduct to be non-unital we obtain a quantum group, that we call a weak C∗Hopf algebra, which is sufficiently general to describe the symmetries of essentially arbitrary fusion rules. This amounts to generalizing the Baaj-Skandalis multiplicative unitaries to multipicative partial isometries. E...

Braided Hopf algebras have attracted much attention in both mathematics and mathematical physics (see e.g. [1][4][13][15][17][16][20][23]). The classification of finite dimensional Hopf algebras is interesting and important for their applications (see [2] [22]). Braided Hopf algebras play an important role in the classification of finite-dimensional pointed Hopf algebras (e.g. [2][1] [19]). The...

Let H be a finite dimensional quasi-Hopf algebra over a field k and A a right H-comodule algebra in the sense of [12]. We first show that on the k-vector space A⊗H∗ we can define an algebra structure, denoted by A # H∗, in the monoidal category of left H-modules (i.e. A # H∗ is an Hmodule algebra in the sense of [2]). Then we will prove that the category of two-sided (A,H)bimodules HM H A is is...

Let H be a finite dimensional non-semisimple Hopf algebra over an algebraically closed field k of characteristic 0. If H has no nontrivial skew-primitive elements, we find some bounds for the dimension of H 1 , the second term in the coradical filtration of H. Using these results, we are able to show that every Hopf algebra of dimension 14, 55 and 77 is semisimple and thus isomorphic to a group...

We give the classification of quiver 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 modules and their corresponding Nichols algebras. In particular, we give the classification of graded Hopf algebras on cotensor coalgebra T c kG(M) of kG-bicomdule M over finite commutative gr...

In [1, 3, 4, 5], a Hopf algebra of rooted trees HR was introduced. It was shown that the antipode of this algebra was the key of a problem of renormalization ([8]). HR is related to the Hopf algebra HCM introduced in [2]. Moreover, the dual algebra of HR is the enveloping algebra of the Lie algebra of rooted trees L. An important problem was to give an explicit construction of the primitive ele...

In this paper we describe Hopf algebras which are associated with certain families of trees. These Hopf algebras originally arose in a natural fashion: one of the authors [5] was investigating data structures based on trees, which could be used to efficiently compute certain differential operators. Given data structures such as trees which can be multiplied, and which act as higherorder derivat...

Any finite-dimensional Hopf algebra has a left and a right integral. Conversely, Larsen and Sweedler showed that, if a finite-dimensional algebra with identity and a comultiplication with counit has a faithful left integral, it has to be a Hopf algebra. In this paper, we generalize this result to possibly infinite-dimensional algebras, with or without identity. We have to leave the setting of H...