On the module structure of free Lie algebras

Abstract

Abstract. We study the free Lie algebra L over a field of non-zero characteristic p as a module for the cyclic group of order p acting on L by cyclically permuting the elements of a free generating set. Our main result is a complete decomposition of L as a direct sum of indecomposable modules.

L. G. Kovács’ Work on Lie Powers

“…The remainder of the paper explains how one can keep track of the indecomposable summands obtained at each degree from this delicate elimination procedure, by using graded modules and formal power series with coefficients in the Green ring. In the case where V is a free KG-module (that is, V is a direct sum of copies of J p ), it is shown that every indecomposable module occurring in the Lie power L n (V) is isomorphic to either J p or J p−1 and explicit formulae are obtained for the multiplicities of these indecomposables (these results were already proved in [4] by a different method). For p ≥ 3, a similar result is shown to hold for the Lie powers L n (J p−1 ), which also consist only of copies of J p and J p−1 , and again explicit formulae for the corresponding multiplicities are obtained.…”

“…This was the topic of Bryant and Stöhr's paper [3] (which set out to answer the above-mentioned question of Laci). This work was subsequently extended by those two authors in [4], where the decomposition problem for Lie powers of a free module for the cyclic group of order p in characteristic p was solved. Of course, the cyclic group of order two can also be considered as the symmetric group of degree two, and it is therefore natural to enquire more generally about the structure of the Lie powers for the natural module for the symmetric group S r .…”

“…The bulk of the paper [19] consists of a thorough examination of Lie powers of the natural GL(2, p)-module in the case p = 3, resulting in the complete solution for the decomposition problem in this case [19,Theorem 6.1]. The central argument built upon previous work of Bryant and Stöhr [4], which dealt with Lie powers of free modules for the group of order p. However, the methods employed in [19] did not immediately carry over to larger primes; the theory needed to be extended and completely new ideas were required. This was accomplished in [8], which conquered the decomposition problem conclusively for Lie powers for groups of prime order.…”

“…The paper [8] concerns Lie powers for cyclic groups of prime order and gives a complete solution of the decomposition problem in this case. The result is highly complex, requiring several clever ideas, many of which laid the groundwork for later [4] progress on modular Lie powers. This article gives a roughly chronological account of the research leading up to the influential paper [8] and some of the developments that followed.…”

Lie powers of free modules for certain groups of prime power order

“…In the case where K has prime characteristic p, and p divides the order of G, it is much more difficult to obtain information about the module structure of L"( V). Recent progress is described in [4] and [5]. The reader is also referred to [4] or [5] for further details of the background and underlying concepts.…”