Nichols algebras associated to the transpositions of the symmetric group are twist-equivalent. (Q2846726)

Let \(V\) be a braided complex vector space, the braiding being given by a solution in \(\text{GL}(V\otimes V)\) satisfying the braid equation, \(\mathfrak B(V)\) the associated Nichols algebra [see \textit{N. Andruskiewitsch} and \textit{H.-J. Schneider}, Math. Sci. Res. Inst. Publ. 43, 1-68 (2002; Zbl 1011.16025)]. \(\mathfrak B(V)\) is a graded Hopf algebra in the category of Yetter-Drinfeld modules over a Hopf algebra \(H\).NEWLINENEWLINE The author studies the case when \(H\) is the group algebra of one of the noncommutative symmetric groups \(S_n\) for \(n>2\). These Nichols algebras are computed from the conjugacy class of transpositions and a 2-cocycle associated to this conjugacy class. The cocycles arise from a cohomology theory defined for racks. For \(n>3\), there are exactly two rack-2 cocycles associated to the conjugacy class of transpositions in \(S_n\) that might have a finite-dimensional Nichols algebra. One is the constant cocycle \(-1\), and the other takes \((s,t)\) to \(1\) if \(s(i)<s(j)\) and to \(-1\) if \(s(j)<s(i)\), where \(t=(i,j)\) with \(i<j\). For each of \(n=4\) and \(n=5\), both Nichols algebras are finite-dimensional and have the same Hilbert series [\textit{G. A. García} and \textit{A. García Iglesias}, Isr. J. Math. 183, 417-444 (2011; Zbl 1231.16023)].NEWLINENEWLINE In the paper under review, the author shows that for \(n>3\), the two rack-2 cocycles are twist equivalent, and thus the two Hilbert series are equal. Twist equivalent means that the two cocycles are related by a form of conjugation involving the rack structure. The proof uses the Schur covering group \(T_n\) of \(S_n\). It is still unknown if the two Nichols algebras associated to \(S_n\) for \(n>5\) are finite-dimensional or not.