The cohomology ring of the 12-dimensional Fomin-Kirillov algebra

Abstract

The 12-dimensional Fomin-Kirillov algebra FK3 is defined as the quadratic algebra with generators a, b and c which satisfy the relations a2=b2=c2=0 and ab+bc+ca=0=ba+cb+ac. By a result of A. Milinski and H.-J. Schneider, this algebra is isomorphic to the Nichols algebra associated to the Yetter-Drinfeld module V, over the symmetric group S3, corresponding to the conjugacy class of all transpositions and the sign representation. Exploiting this identification, we compute the cohomology ring ExtFK3*(k,k), showing that it is a polynomial ring S[X] with coefficients in the symmetric braided algebra of V. As an application we also compute the cohomology rings of the bosonization FK3#kS3 and of its dual, which are 72-dimensional ordinary Hopf algebras.