Bicovariant differential calculi on SL q(N) and Sp q(N)

For transcendental values of the deformation parameter q all bicovariant first order differential calculi on the Hopf algebras are classified.     

Yetter–Drinfeld modules over bosonizations of dually paired Hopf algebras

Let (R^∨,R)$(R^{\vee },R)$ be a dual pair of Hopf algebras in the category of Yetter–Drinfeld modules over a Hopf algebra H$H$ with bijective antipode. We show that there is a braided monoidal isomorphism between rational left Yetter–Drinfeld modules over the bosonizations of R$R$ and of R^∨$R^{\vee }$, respectively. As an application of this very general category isomorphism we obtain a natural proof of the existence of reflections of Nichols algebras of semi-simple Yetter–Drinfeld modules over H$H$.     

Nichols algebras of group type with many quadratic relations

We classify Nichols algebras of irreducible Yetter–Drinfeld modules over nonabelian groups satisfying an inequality for the dimension of the homogeneous subspace of degree two. All such Nichols algebras are finite-dimensional, and all known finite-dimensional Nichols algebras of nonabelian group type appear in the result of our classification. We find a new finite-dimensional Nichols algebra over fields of characteristic two.     

Classification of arithmetic root systems

Arithmetic root systems are invariants of Nichols algebras of diagonal type with a certain finiteness property. They can also be considered as generalizations of ordinary root systems with rich structure and many new examples. On the other hand, Nichols algebras are fundamental objects in the construction of quantized enveloping algebras, in the noncommutative differential geometry of quantum groups, and in the classification of pointed Hopf algebras by the lifting method of Andruskiewitsch and Schneider. In the present paper arithmetic root systems are classified in full generality. As a byproduct many new finite dimensional pointed Hopf algebras are obtained.