On the computation of Hopf 2-cocycles with an example of diagonal type

Abstract We present a framework for the computation of Hopf 2-cocycles involved in deformations Nichols algebras over semisimple algebras. write down recurrence formula and investigate extent connection with invariant Hochschild cohomology terms exponentials. As an example, we detailed computations leading to explicit description algebra Cartan type $A_2$ $q=-1$ , a.k.a. positive part small quantum group $\mathfrak{u}^+_{\sqrt{-\text{1}}}(\mathfrak{sl}_3)$ . show that these cocycles are generically pure is they not cohomologous exponentials 2-cocycles.