Lie identities for Hopf algebras
Let \(k\) be a field of characteristic \(p>0\). If \(R\) is a \(k\)-algebra then we set \(\gamma_0(R)=R\) and inductively \(\gamma_{n+1}(R)\) is the linear span of all commutators \([x,y]=xy-yx\), where \(x\in\gamma_n(R)\), \(y\in R\). Suppose that \(p>2\) and \(R=u(L)\) is the restricted universal envelope of the Lie \(p\)-algebra \(L\). If \([[\gamma_n(R),\gamma_r(R)],R]=0\) for some \(n,r\geq 1\), then \(L\) is a nilpotent Lie algebra and the ring \(R\) is Lie-nilpotent. Let \(R\) be the group ring \(kG\) of a group \(G\) and either \(p>3\) and \([[\gamma_n(R),\gamma_r(R)],R]=0\) or \(p=3\) and \([\gamma_n(R),\gamma_r(R)]=0\) for some \(n,r\geq 1\). Then \(G\) is a nilpotent group and the ring \(R\) is Lie-nilpotent. Note that a citerion of Lie nilpotence of \(u(L)\) was found by \textit{D. M. Riley} and \textit{A. Shalev} [J. Algebra 162, No. 1, 46-61 (1993; Zbl 0798.17011)]. A similar result for group rings was proved by \textit{I. B. S. Passi, D. S. Passman} and \textit{S. K. Sehgal} [Can. J. Math. 25, 748-757 (1973; Zbl 0266.16011)]. Related problems are discussed in the reviewer's paper [Contemp. Math. 184, 33-42 (1995; Zbl 0836.16019)].
scientific article; zbMATH DE number 1091340