CONICET | Buscador de Institutos y Recursos Humanos

The 12-dimensional Fomin?Kirillov algebra FK_3 is defined as the quadratic algebra with generators a, b and c which satisfy the relations a^2=b^2=c^2=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 S_3, corresponding to the conjugacy class of all transpositions and the sign representation. Exploiting this identification, we compute the cohomology ring Ext_FK3(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 FK_3#kS_3 and of its dual, which are 72-dimensional ordinary Hopf algebras