This paper introduces five notions, including π-algebras, π-ideals, Hopf π-algebras, π-modules and Hopf π-modules, verifies the fundamental isomorphism theorem of π-algebras and studies some algebraic properties of Hopf π-algebras as well.

Hopf π- Algebras

Groupes et algébres de lie

Concrete mathematics, a foundation for computer science, by Ronald L. Graham, Donald E. Knuth and Oren Patashnik. Pp 625. £24·95. 1989. ISBN 0-201-14236-8 (Addison-Wesley)

We classify finite-dimensional complex Hopf algebras A which are pointed, that is, all of whose irreducible comodules are one-dimensional, and whose group of group-like elementsG.A/ is abelian such that all prime divisors of the order of G.A/ are> 7. Since these Hopf algebras turn out to be deformations of a natural class of generalized small quantum groups, our result can be read as an axiomatic description of generalized small quantum groups.

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On the classification of finite-dimensional pointed Hopf algebras

Classification of arithmetic root systems

The theory of Nichols algebras of diagonal type is known to be closely related to that of semi-simple Lie algebras. In this paper the connection between both theories is made closer. For any Nichols algebra of diagonal type invertible transformations are introduced, which remind one of the action of the Weyl group on the root system associated to a semi-simple Lie algebra. They give rise to the definition of a groupoid. As an application an alternative proof of classification results of Rosso, Andruskiewitsch, and Schneider is obtained without using any technical assumptions on the braiding.

The Weyl groupoid of a Nichols algebra of diagonal type

We study the Nichols algebra of a semisimple Yetter-Drinfeld module and introduce new invariants including the notions of real roots and the Weyl groupoid. The crucial ingredient is a "reflection" defined on arbitrary such Nichols algebras. Our construction generalizes the restriction of Lusztig's automorphisms of quantized Kac-Moody algebras to the nilpotent part. As a direct application we complete the classifications of finite-dimensional pointed Hopf algebras over ${\Bbb S}_3$, and of finite-dimensional Nichols algebras over ${\Bbb S}_4$. This theory has led to surprising new results in the classification of finite-dimensional pointed Hopf algebras with a non-abelian group of group-like elements.

The Nichols algebra of a semisimple Yetter-Drinfeld module

We study the Nichols algebra of a semisimple Yetter-Drinfeld module and introduce new invariants such as real roots. The crucial ingredient is a `reflection' in the class of such Nichols algebras. We conclude the classifications of finite-dimensional pointed Hopf algebras over S_3, and of finite-dimensional Nichols algebras over S_4. 
The revised version contains an extended introduction with references to recent applications, and a simplified definition of the Weyl groupoid of a semisimple Yetter-Drinfeld module. 
Key words: Hopf algebras, quantum groups, Weyl groupoid

Motivated by work of Kac and Lusztig, we define a root system and a Weyl groupoid for a large class of semisimple Yetter-Drinfeld modules over an arbitrary Hopf algebra. The ob- tained combinatorial structure fits perfectly into an existing frame- work of generalized root systems associated to a family of Cartan matrices, and provides novel insight into Nichols algebras. We demonstrate the power of our construction with new results on Nichols algebras over finite non-abelian simple groups and sym- metric groups.

István Heckenberger

Papers

Classification of arithmetic root systems

The Weyl groupoid of a Nichols algebra of diagonal type

The Nichols algebra of a semisimple Yetter-Drinfeld module

The Nichols algebra of a semisimple Yetter-Drinfeld module

Root systems and Weyl groupoids for Nichols algebras