Some Properties and Examples of Triangular Pointed Hopf Algebras

We extend to the co-Frobenius case a result of Drinfeld and Radford related to the fourth power of the antipode of a finite dimensional (co)quasitriangular Hopf algebra.

We parameterize the finite-dimensional irreducible representations of a class of pointed Hopf algebras over an algebraically closed field of characteristic zero by dominant characters. The Hopf algebras we are considering arise in the work of N. Andruskiewitsch and the second author. Special cases are the multiparameter deformations of the enveloping algebras of semisimple Lie algebras where the deforming parameters are not roots of unity and some of their finite-dimensional ...

Let p and q be distinct odd primes and assume k is an algebraically closed field of characteristic zero. We classify all quasitriangular Hopf algebras of dimension pq^2 over k, which are not simple as Hopf algebras. Moreover, we obtained all quasitriangular structures on these Hopf algebras.

Actions of semisimple Hopf algebras H over an algebraically closed field of characteristic zero on commutative domains were classified recently by the authors. The answer turns out to be very simple- if the action is inner faithful, then H has to be a group algebra. The present article contributes to the non-semisimple case, which is much more complicated. Namely, we study actions of finite dimensional (not necessarily semisimple) Hopf algebras on commutative domains, particu...

Let $U$ and $A$ be algebras over a field $k$. We study algebra structures $H$ on the underlying tensor product $U{\otimes}A$ of vector spaces which satisfy $(u{\otimes}a)(u'{\otimes}a') = uu'{\otimes}aa'$ if $a = 1$ or $u' = 1$. For a pair of characters $\rho \in \Alg(U, k)$ and $\chi \in \Alg(A, k)$ we define a left $H$-module $L(\rho, \chi)$. Under reasonable hypotheses the correspondence $(\rho, \chi) \mapsto L(\rho, \chi)$ determines a bijection between character pairs an...

This is a continuation of the authors' study of finite-dimensional pointed Hopf algebras H which act inner faithfully on commutative domains. As mentioned in Part I of this work, the study boils down to the case where H acts inner faithfully on a field. These Hopf algebras are referred to as Galois-theoretical. In this work, we provide classification results for finite-dimensional pointed Galois-theoretical Hopf algebras H of finite Cartan type. Namely, we determine when su...

A basis B of a finite dimensional Hopf algebra H is said to be positive if all the structure constants of H relative to B are non-negative. A quasi-triangular structure $R\in H\otimes H$ is said to be positive with respect to B if it has non-negative coefficients in the basis $B \otimes B$ of $H\otimes H$. In our earlier work, we have classified all finite dimensional Hopf algebras with positive bases. In this paper, we classify positive quasi-triangular structures on such Ho...

A class of finite-dimensional Hopf algebras which generalise the notion of Taft algebras is studied. We give necessary and sufficient conditions for these Hopf algebras to omit a pair in involution, that is, to not have a group-like and a character implementing the square of the antipode. As a consequence we prove the existence of an infinite set of examples of finite-dimensional Hopf algebras without such pairs. This has implications for the theory of anti-Yetter-Drinfeld mo...

Let $p,q$ be prime numbers with $p>q^3$, and $k$ an algebraically closed field of characteristic 0. In this paper, we obtain the structure theorems for semisimple Hopf algebras of dimension $pq^3$.

Let k be an algebraically closed field of characteristic 0 and let D_m be the dihedral group of order 2m with m= 4t, with t bigger than 2. We classify all finite-dimensional Nichols algebras over D_m and all finite-dimensional pointed Hopf algebras whose group of group-likes is D_m, by means of the lifting method. Our main result gives an infinite family of non-abelian groups where the classification of finite-dimensional pointed Hopf algebras is completed. Moreover, it provi...