2005/06/06 by R. M. Bryant, Bryant, R. M., Manfred Schocker +2
Mathematics · #17B01 #20C07 #20C20 #Advanced Topics in Algebra #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT) #Rings and Algebras (math.RA) #math.RA #math.RT #msc:17B01 #msc:20C07 #msc:20C20
paper · pdf · doi:10.48550/arxiv.math/0506104
20 pages
arxiv created 2005/06/06 · openalex publication_date 2005/06/06 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let G be a group, F a field of prime characteristic p and V a finite-dimensional FG-module. Let L(V) denote the free Lie algebra on V, regarded as an FG-module, and, for each positive integer r, let Lr(V) be the rth homogeneous component of L(V), called the rth Lie power of V. In a previous paper we obtained a decomposition of Lr(V) as a direct sum of modules of the form Ls(W), where s is a power of p. Here we derive some consequences. First we obtain a similar result for restricted Lie powers of V. Then we consider the `Lie resolvents' Φr : certain functions on the Green ring of FG which determine Lie powers up to isomorphism. For k not divisible by p, we obtain the factorisation Φpmk = Φpm ∘ Φk, separating out the key case of p-power degree. Finally we study certain functions on power series over the Green ring, denoted by \bf S^* and \bf L^*, which encode symmetric powers and Lie powers, respectively. In characteristic 0, \bf L^* is the inverse of \bf S^*. In characteristic p, the composite \bf L^* ∘ \bf S^* maps any p-typical power series to a p-typical power series.