Enshrining Finite Simple Groups

Regents Of The University Of Michigan - Ann Arbor, Ann Arbor MI

Investigators

Abstract

The PI will work on problems in finite simple groups, lattices and vertex operator algebras (VOAs). Work on VOAs is joint with Chongying Dong. The main goal is to build moonshine-like VOAs for finite simple groups (as done for the monster in the mid 80s). Lattice constructions and analyses support this goal but recent work on lattices has a life of its own due to new techniques and applications. (1) The PI plans to strengthen recent joint work on uniqueness for the important moonshine VOA by reducing hypotheses. (2) The PI plans to give new constructions of sporadic groups and VOAs which enshrine them. This will involve revisions of existing moonshine VOA theories and adaptions to particular sporadic groups, and other work with the more classic lattice type VOAs. The result would be a new and uniform setting of most finite simple groups. (3) The PI's recent work on lattices has concentrated on spinoffs of the Barnes-Wall series. The PI will continue this study. The PI will build new series of lattices with series of finite groups as automorphism groups. Certain of these series and spinoffs will be used in part (2). Some of these lattices have relatively high minimum norms (and could be extremal), so the PI will try to settle those norms. Automorphism groups of some of these lattices will be determined. Uniqueness theories will be developed. (4) The PI plans to do more work on automorphism groups of low rank VOAs and on the connections between nonassociative algebras and VOAs. The PI hopes this proposal will help integrate sporadic groups into traditionally mainstream mathematics. Sporadic simple groups are finite simple groups which are not naturally part of the infinite series (classical groups, alternating groups). The program with lattices links certain infinite series of lattices involving classical groups over finite fields with those associated to sporadic groups. The program with VOA theory will link finite groups, general algebraic groups and infinite dimensional Lie theory. Lattices, VOAs and simple groups are connected to many parts of mathematics and mathematical physics.