The Nichols algebra of a semisimple Yetter-Drinfeld module

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.