## Infinite-dimensional Lie algebras

### Dr. Igor Mencattini

Contents:

The Virasoro algebra

- Highest weight and vaema modules
- Oscillator (Heisenberg) algebra: Segal-Sugawara construction
- Unitarity and irreducibility
- Kac's Determinant formula

Kac-Moody algebras (of the affine type)

- Root systems and Weyl groups
- classification of affine Kac-Moody algebras
- Highest weight theory (non abelian Segal-Sugawara construction)
- Character formula (MacDonald identities)

Applications: algebras of infinite matrices, boson-fermion correspondence, application to soliton equations

References:

- Kac-Raina: Highest weight representation of infinite-dimensional Lie algebras