Nichols Algebras over finite simple groups

Given a pair consisting of a vector space V and a solution of the braid equation c\in End(V\otimes V), one defines a graded algebra, called the associated Nichols algebra B(V,c). Notable examples of Nichols algebras are: the symmetric algebra, the exterior algebra, the positive part of quantized enveloping algebras, the Fomin-Kirillov algebras of degree 3,4,5 and conjecturally all Fomin-Kirillov algebras. Such algebras are crucial in the classification program of finite-dimensional pointed Hopf algebras, and lead to the natural question: for which pairs (V,c) is B(V,c) finite-dimensional? When (V,c) is obtained starting from a non-abelian finite simple group G and a representation of G with compatible G-grading (i.e., a Yetter-Drinfeld module for G), it has been conjectured that B(V,c) is never finite-dimensional. In this talk, based on joint work with N. Andruskiewitsch, I will describe the state-of-the-art of this conjecture.