Restricted Lazard elimination and modular Lie powers (Q2765557)

Let \(L(Y)\) denote the free Lie algebra on a set \(Y\) over a commutative ring \(K\). Whenever \(Y\) is written as the disjoint union of two subsets \(Y_1\) and \(Y_2\), the free Lie algebra \(L(Y)\) decomposes as a direct sum of its free Lie subalgebra \(L(Y_1)\) and the ideal generated by \(Y_2\). Furthermore, the latter is itself a free Lie algebra, and a free generating set for it can be explicitly described in terms of \(Y_1\) and \(Y_2\). This basic result is referred to as Lazard elimination (of the subalgebra \(L(Y_1)\)). The special case where \(Y_1=\{x\}\) is a singleton is already sufficient for many applications, among which, the constructions of Hall bases of free Lie algebras of finite rank. In this case, the direct decomposition becomes \(L(Y)=\langle x\rangle\oplus L(Y| x)\), where \(Y| x=\{[y,x^k];\;y\in Y\setminus\{x\},\;k\geq 0\}\) and \([y,x^k]=[y,x,\ldots,x]\) (with \(x\) occurring \(k\) times, and with the left-normed convention for Lie brackets).NEWLINENEWLINEThe paper under review presents two applications of a variation of Lazard elimination for free restricted Lie algebras over a field \(K\) of characteristic \(p\). If \(x\) is an element of the set \(Y\), the free restricted Lie algebra \(R(Y)\) on \(Y\) over \(K\) decomposes into a direct sum of its subspace \(\langle x\rangle\) and the ideal generated by \(x^p\) and \(Y\setminus\{x\}\). The latter is itself a free restricted Lie algebra with free generating set \(Y| _r\,x=\{x^p,\;[z,x^\alpha];\;z\in Y\setminus\{x\},\;0\leq\alpha<p\}\). This result is referred to as restricted elimination of the free generator \(x\).NEWLINENEWLINEThe first application of restricted elimination presented in the paper concerns free Lie algebras over a field \(K\) of characteristic two. If \(G\) is the group of order two and \(V\) is a finite-dimensional free \(KG\)-module, the author computes an explicit decomposition of the Lie and restricted Lie powers of \(V\) into indecomposables. Apart from a minor improvement this was already known from a previous paper of \textit{S. Guilfoyle} and \textit{L. Stöhr} [J. Algebra 204, 337--346 (1998; Zbl 0977.17003)], but the present proof based on restricted Lazard elimination is much simpler.NEWLINENEWLINEThe second application is more involved and is a technical result on Lie powers of indecomposable modules for the holomorph of the group \(P\) of order \(p\) (over a field \(K\) of characteristic \(p\)). This extends an analogous result obtained for \(P\) in [\textit{R. M. Bryant}, \textit{L. G. Kovacs} and \textit{R. Stöhr}, Proc. Lond. Math Soc. (3) 84, 343--374 (2002; Zbl 1021.20008)]. The present form of the result is better suited to the investigation of the Lie powers of the natural module for \(\text{ GL}(2,p)\) which is pursued in [\textit{R. M. Bryant}, \textit{L. G. Kovacs} and \textit{R. Stöhr}, J. Algebra 260, 617--630 (2003; Zbl 1130.20309)]. However, the author emphasizes that even more important than this extension is the use of restricted Lazard elimination made in the proof.