\titleofpresentation{Necklaces and free Lie algebras}

\presenter{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 $B_n$.

\presenteraddress{University of Manchester}

\email{roger.bryant@manchester.ac.uk}
