Related papers: On Nichols algebras over basic Hopf algebras
We classify all non-affine Hopf algebras $H$ over an algebraically closed field $k$ of characteristic zero that are integral domains of Gelfand-Kirillov dimension two and satisfy the condition $\text{Ext}^1_H(k, k) \neq 0$. The affine ones…
We introduce a new filtration on Hopf algebras, the standard filtration, generalizing the coradical filtration. Its zeroth term, called the Hopf coradical, is the subalgebra generated by the coradical. We give a structure theorem: any Hopf…
We give an explicit and essentially minimal list of defining relations of a Nichols algebra of diagonal type with finite root system. This list contains the well-known quantum Serre relations but also many new variations. A conjecture by…
We translate the concept of restriction of an arrangement in terms of Hopf algebras. In consequence, every Nichols algebra gives rise to a simplicial complex decorated by Nichols algebras with restricted root systems. As applications, some…
We compute the finite-dimensional Nichols algebras over the sum of two simple Yetter-Drinfeld modules V and W over non-abelian quotients of a certain central extension of the dihedral group of eight elements or SL(2,3), and such that the…
Let $H$ be a pointed Hopf algebra over an algebraically closed field of characteristic zero. If $H$ is a domain with finite Gelfand-Kirillov dimension greater than or equal to two, then $H$ contains a Hopf subalgebra of Gelfand-Kirillov…
Nichols algebras are fundamental objects in the construction of quantized enveloping algebras and in the classification of pointed Hopf algebras by lifting method of Andruskiewitsch and Schneider. Arithmetic root systems are invariants of…
We study the pointed or copointed liftings of Nichols algebras associated to affine racks and constant cocycles for any finite group admitting a principal YD-realization of these racks. In the copointed case we complete the classification…
It is shown that Nichols algebras over alternating groups A_m, m>4, are infinite dimensional. This proves that any complex finite dimensional pointed Hopf algebra with group of group-likes isomorphic to A_m is isomorphic to the group…
We generalize various properties of Yetter-Drinfeld modules over Hopf algebras to quasi-Hopf algebras. The dual of a finite dimensional Yetter-Drinfeld module is again a Yetter-Drinfeld module. The algebra $H_0$ in the category of…
The method of subquotients is developed and used to determine all finite dimensional rank 2 Nichols algebras of diagonal type over an arbitrary field of characteristic zero. Key Words: Hopf algebra, Nichols algebra
We study finite dimensional representations over some Noetherian algebras over a field of characteristic zero. More precisely, we give necessary and sufficient conditions for the category of locally finite dimensional representations to be…
This is a report on the present state of the problem of determining the dimension of the Nichols algebra associated to a rack and a cocycle. This is relevant for the classification of finite-dimensional complex pointed Hopf algebras whose…
It is shown in math.QA/0301027 that a finite dimensional quasi-Hopf algebra with radical of codimension 1 is semisimple and 1-dimensional. On the other hand, there exist quasi-Hopf (in fact, Hopf) algebras, whose radical has codimension 2.…
A fundamental problem in the theory of Hopf algebras is the classification and explicit construction of finite-dimensional quasitriangular Hopf algebras over C. These Hopf algebras constitute a very important class of Hopf algebras,…
We classify all noetherian Hopf algebras $H$ over an algebraically closed field $k$ of characteristic zero which are integral domains of Gelfand-Kirillov dimension two and satisfy the condition $\Ext^1_H(k,k)\neq 0$. The latter condition is…
Let $H$ be a pointed Hopf algebra. We show that under some mild assumptions $H$ and its associated graded Hopf algebra $\gr H$ have the same Gelfand-Kirillov dimension. As an application, we prove that the Gelfand-Kirillov dimension of a…
Let $H$ be a finite-dimensional Hopf algebra over an algebraically closed field of characteristic 0. If $H$ is not semisimple and $\dim(H)=2n$ for some odd integer $n$, then $H$ or $H^*$ is not unimodular. Using this result, we prove that…
We show that all possible categories of Yetter-Drinfeld modules over a quasi-Hopf algebra $H$ are isomorphic. We prove also that the category $\yd^{\rm fd}$ of finite dimensional left Yetter-Drinfeld modules is rigid and then we compute…
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…