# Natalya Dobridneva

## Some sublattices of the Leech lattice and applications

We determine the orbits of fixed-point sublattices of the Leech lattice with respect to the action of the Conway group $Co$. The Leech lattice $\Lambda$ is the unique positive definite, even, unimodular lattice of rank~$24$ without roots. It may also be characterized as the most densely packed lattice in dimension $24$. The group of isometries of $\Lambda$ is the Conway group $Co$. For a subgroup $H\subseteq Co$ we set $\Lambda^H = \{ v\in\Lambda | hv=v \mbox{\ for all\ } h\in H\}.$ We call such a sublattice of $\Lambda$ a fixed-point sublattice. Let $\mathcal{F}$ be the set of all fixed-point sublattices of $\Lambda$. The Conway group acts by translation on $\mathcal{F}$, because if $g\in Co$, then $g \Lambda^G = \Lambda^{gHg^{-1}}.$ In this talk, we classify the $Co$-orbits of fixed-point sublattices.

The purpose of the present talk is not merely to enumerate the orbits of fixed-point sublattices, but to provide in addition a detailed analysis of their properties. The Leech lattice is also the starting point of the construction of interesting vertex operator algebras and generalized Kac-Moody Lie algebras. Such Kac-Moody Lie algebras have root lattices that can often be described in terms of fixed-point lattices inside $\Lambda$, and the associated denominator identities provide Moonshine for the corresponding subgroups.