We propose the following principle to study pointed Hopf algebras, or more generally, Hopf algebras whose coradical is a Hopf subalgebra. Given such a Hopf algebra A, consider its coradical filtration and the associated graded coalgebra grA. Then grA is a graded Hopf algebra, since the coradical A0 of A is a Hopf subalgebra. In addition, there is a projection π : grA → A0; let R be the algebra ...

We construct a three parameter deformation of the Hopf algebra LDIAG. This new algebra is a true Hopf deformation which reduces to LDIAG on one hand and to MQSym on the other, relating LDIAG to other Hopf algebras of interest in contemporary physics. Further, its product law reproduces that of the algebra of polyzeta functions.

We study a natural construction of Hopf algebra quotients canonically associated to an R-matrix in a finite dimensional Hopf algebra. We apply this construction to show that a quasitriangular Hopf algebra whose dimension is odd and square-free is necessarily cocommutative.

All −1-type pointed Hopf algebras with Weyl groups of exceptional type are found. It is proved that every non −1-type pointed Hopf algebra is infinite dimensional. 2000 Mathematics Subject Classification: 16W30, 16G10 keywords: Quiver, Hopf algebra, Weyl group.

We construct a three parameter deformation of the Hopf algebra LDIAG. This new algebra is a true Hopf deformation which reduces to LDIAG on one hand and to MQSym on the other , relating LDIAG to other Hopf algebras of interest in contemporary physics. Further , its product law reproduces that of the algebra of polyzeta functions .

The invariants of 3-manifolds defined by Kuperberg for involutory Hopf algebras and those defined by the authors for spherical Hopf algebras are the same for Hopf algebras on which they are both defined. Introduction The purpose of this paper is to compare two previously defined invariants of 3-manifolds. Let A be a finite-dimensional Hopf algebra over a field F with antipode S. Then if S = 1 t...

Hopf-Galois extensions are known to be the right generalizations of both Galois field extensions and principal G-bundles in the framework of non-commutative associative algebras. An abundant literature has been devoted to them by Hopf algebra specialists (see [Mg], [Sn1], [Sn2] and references therein). Recently there has been a surge of interest in Hopf-Galois extensions among mathematicians an...

The graph algebra is a commutative, cocommutative, graded, connected incidence Hopf algebra, whose basis elements correspond to finite graphs and whose Hopf product and coproduct admit simple combinatorial descriptions. We give a new formula for the antipode in the graph algebra in terms of acyclic orientations; our formula contains many fewer terms than Takeuchi’s and Schmitt’s more general fo...

This exposition concerns two different notions of Frobenius-Schur indicators for finite-dimensional Hopf algebras. These two versions of indicators coincide when the underlying Hopf algebra is semisimple. We are particularly interested in the family of pivotal finite-dimensional Hopf algebras with unique pivotal element; both indicators are gauge invariants of this family of Hopf algebras. We o...

In this paper, we construct explicitly a noncommutative symmetric (NCS) system over the Grossman-Larson Hopf algebra of labeled rooted trees. By the universal property of the NCS system formed by the generating functions of certain noncommutative symmetric functions, we obtain a specialization of noncommutative symmetric functions by labeled rooted trees. Taking the graded duals, we also get a ...