Infinite-dimensional algebraic and splittable Lie algebras

Infinite-dimensional Lie algebras

Dual pairs and infinite dimensional Lie algebras

We construct and study various dual pairs between finite dimensional classical Lie groups and infinite dimensional Lie algebras in some Fock representations. The infinite dimensional Lie algebras here can be either a completed infinite rank affine Lie algebra, the W1+∞ algebra or its certain Lie subalgebras. We give a formulation in the framework of vertex algebras. We also formulate several co...

Nongraded Infinite-Dimensional Simple Lie Algebras

Nongraded infinite-dimensional Lie algebras appeared naturally in the theory of Hamiltonian operators, the theory of vertex algebras and their multi-variable analogues. They play important roles in mathematical physics. This survey article is written based on the author’s seminar talks on nongraded infinite-dimensional simple Lie algebras. The key constructional ingredients of our Lie algebras ...

Math 263: Infinite-Dimensional Lie Algebras

Our text will be Kac, Infinite-Dimensional Lie Algebras (third edition) though I will also rely on Kac and Raina, Bombay Lectures on Highest Weight Representations of InfiniteDimensional Lie Algebras . I will assume you have access to the first text but not the second. Unfortunately Kac’ book is long and dense. These notes are an attempt to make a speedy entry. Most relevant for the first part ...

Infinite-Dimensional Lie Groups and Algebras in Mathematical Physics

We give a review of infinite-dimensional Lie groups and algebras and show some applications and examples in mathematical physics. This includes diffeomorphism groups and their natural subgroups like volume-preserving and symplectic transformations, as well as gauge groups and loop groups. Applications include fluid dynamics, Maxwell’s equations, and plasma physics. We discuss applications in qu...