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.