From Hamiltonian mechanics to homotopical Lie theory
Vortrag von Dr. Chris Rogers Datum: 10.11.14 Zeit: 16.15 - 17.15 Raum: Y27H28
In Hamiltonian mechanics, we model the phase space of a physical system using symplectic geometry, and we use Lie algebras to describe the space's infinitesimal symmetries. Given such a Lie algebra of symmetries, the geometry naturally produces for us a new Lie algebra called a "central extension". This central extension plays a crucial role, especially in quantization. The famous Heisenberg algebra, for example, arises precisely in this way.
In this talk, I will explain how the above recipe can be enhanced to geometrically produce examples of what are called "homotopy Lie algebras". These are chain complexes equipped with structures which satisfy the axioms of a Lie algebra only up to chain homotopy. They provide important tools for rational homotopy theory and deformation quantization. The homotopy Lie algebras produced from our construction turn out to have beautiful relationships with the theory of loop groups, and what are called "string structures" in algebraic topology.