Lie powers of the natural module for GL(2)

Research Report MRR98-035

Lie powers of the natural module for GL(2)

L.G. Kovács and Ralph Stöhr

Abstract: Let L be a free Lie algebra of finite rank r over a field F. For each positive integer n, denote by

the degree n homogeneous component of L. The group of graded algebra automorphisms of L may be identified with $GL(r,\F)$ in such a way that

becomes the natural module, and then the

are called the Lie powers of this module.

This paper is concerned with the smallest case, when r=2 and F is the prime field of p elements. The isomorphism types of the indecomposable GL(2,p)-modules are known, and the problem is to give the (Krull-Schmidt) multiplicities of these indecomposables in the unrefinable direct decompositions of the . Conclusive answers are obtained when either $p\leq3$ or n is prime to p.

The analogous question about the free Lie agebra of rank 2 over Z is also answered, not for $GL(2,\Z)$ but for every maximal finite subgroup of $GL(2,\Z)$ .

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