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

The $12$-dimensional Fomin-Kirillov algebra $F{K}_3$ 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 ${\mathbb{S}}_3$, corresponding to the conjugacy class of all transpositions and the sign representation. Exploiting this identification, we compute the cohomology ring $Ex{t}_{F{K}_3}^{\ast }(\mathbb{k},\mathbb{k})$, 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 $F{K}_3\mathrm{\#}\mathbb{k}{\mathbb{S}}_3$ and of its dual, which are $72$-dimensional ordinary Hopf algebras.