Introduction
Infinite dimensional Lie algebras allow structures over infinite bases that capture symmetries in infinite-parameter systems, such as those arising in conformal field theory, integrable systems, and quantum physics. Key examples include the Witt algebra of vector fields on the circle, its central extension known as the Virasoro algebra, the Heisenberg algebra, and Kac-Moody algebras, which generalize finite-dimensional semisimple Lie algebras via generalized Cartan matrices and Dynkin diagrams, enabling classifications beyond the Cartan-Killing theorem.
Representation theory in this context focuses on highest-weight modules, Verma modules, and characters, with Weyl’s complete reducibility theorem failing in general, leading to indecomposable yet irreducible quotients and applications in modular forms and string theory. Unlike finite-dimensional cases, infinite-dimensional Lie algebras often require locally convex topologies for Lie groups, complicating the exponential map and integrability from algebras to groups, though special classes like affine Kac-Moody algebras admit global realizations.
Contents
Heisenberg Algebra and Fock Models
Witt Algebra and Virasoro Algebra
The Verma Module
Unitary Highest Weight Representations
The Integrability Theorem
References
Bombay Lectures on Highest Weight Representations of Infinite Dimensional Lie Algebras