Representations in Infinite-Dimensional Lie Theory

点击翻译按钮获取中文摘要

Lie algebras are mathematical structures that describe symmetries of certain physical systems. Many such systems have infinitely many independent symmetries, so it is natural to study Lie algebras which are infinite-dimensional. For example, some of the most interesting infinite-dimensional Lie algebras, called affine Kac-Moody Lie algebras, describe symmetries in string theory.******Unlike other infinite-dimensional Kac-Moody algebras, the affine ones are extensions of spaces of functions from the circle to simple finite-dimensional Lie algebras. This geometric interpretation is a large part of what makes affine Lie algebras so special among all other Kac-Moody algebras. It has led to spectacular interactions with many areas of pure mathematics and particle physics, such as vertex operator algebras, integrable systems, quantum groups, knot invariants, modular forms, and conformal field theory.******In the last few years, mathematicians have begun studying other Lie algebras of this type, called current algebras, based on algebras of functions from more general spaces, called affine schemes, to finite-dimensional Lie algebras. Much less is known about these algebras, though they appear very naturally in both physics and pure mathematics because of the beautiful ways in which they act as symmetries of geometric spaces, called representations. ******After determining which kinds of symmetries are the most important, I plan to classify these representations of current algebras and their subalgebras left invariant by groups of transformations. Together with other experts, I will also explore a complex relationship called Kazhdan-Lusztig equivalence between the classical and quantum world, and the reconstruction of current algebras from their representation theory. My students will work with related symmetry algebras, called toroidal algebras, superconformal algebras, and affine W-algebras. Each of these is defined in terms of current algebras: toroidal algebras come from functions on a torus (a mathematical doughnut), superconformal algebras are described by data attached to current algebras of conformal superalgebras, and affine W-algebras are constructed from affine Lie algebras via a process called quantum hamiltonian reduction.******The research program should lead to a better understanding of what current algebras are, and how they act as symmetries. It should also enhance our knowledge of affine Lie algebras through new techniques inspired by quantum groups. The student projects will make important links with vertex operator algebras, which can be viewed as algebraic analogues of conformal field theory. *****