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Necklaces and free Lie algebras



Roger Bryant



I shall first talk about the well-known formula (Witt's formula) that

gives the number of primitive necklaces of length $n$ and the dimension

of the $n$th homogeneous component of a free Lie algebra. I shall then

describe some recent results concerned with the module structure

of a free Lie algebra under the action of a group. Let $G$ be a group, $K$

a field and $V$ a finite-dimensional $KG$-module. Let $L(V)$ be the free

Lie algebra over $K$ that has $V$ as a subspace and every basis of $V$ as a

free generating set. The action of each element of $G$ on $V$ extends to a

Lie algebra automorphism of $L(V)$. Thus $L(V)$ becomes a $KG$-module, and

each homogeneous component $L^n(V)$ is a $KG$-submodule called the $n$th

Lie power of $V$. I shall describe joint work with Manfred Schocker that

gives a general decomposition theorem for Lie powers in terms of modules

denoted by $B_n$. I shall also report on recent work with Marianne

Johnson that clarifies the structure of these modules $B_n$.



University of Manchester



roger.bryant@manchester.ac.uk





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