Factorisation of Lie resolvents (Q860433)

Let \(G\) be a group and \(F\) a field of characteristic \(p>0\). For any finite-dimensional \(FG\)-module \(V\), let \(L(V)\) be the free Lie algebra on V, and let \(L^r(V)\) be the \(r\)th homogeneous component of \(L(V)\). \(L^r(V)\) is called the \(r\)th Lie power of \(V\). Likewise the homogeneous component \(R^r(V)\) of the free restricted Lie algebra \(R(V)\) is called the \(r\)th restricted Lie power of \(V\). The paper under review is a further contribution of the authors to studying the modules \(L^r(V)\) and \(R^r(V)\) up to isomorphism. Its main result is a factorisation theorem for maps \(\Phi^r:R_{FG} \rightarrow R_{FG}\) known as Lie resolvents which, according to an earlier paper by the first author [J. Lond. Math. Soc., II. Ser. 68, No. 2, 355--370 (2003; Zbl 1056.17002)], are crucial for understanding isomorphism classes of Lie powers. It states that if \(p\) does not divide \(k\), then \(\Phi^{p^mk}= \Phi^{p^m}\circ\Phi^k\). This essentially reduces the problem of understanding Lie resolvents to the case of \(p\)-power degree.