Book contents
- Frontmatter
- Contents
- 1 Introduction
- 2 Integrable dynamical systems
- 3 Synopsis of integrable systems
- 4 Algebraic methods
- 5 Analytical methods
- 6 The closed Toda chain
- 7 The Calogero—Moser model
- 8 Isomonodromic deformations
- 9 Grassmannian and integrable hierarchies
- 10 The KP hierarchy
- 11 The KdV hierarchy
- 12 The Toda field theories
- 13 Classical inverse scattering method
- 14 Symplectic geometry
- 15 Riemann surfaces
- 16 Lie algebras
- Index
16 - Lie algebras
Published online by Cambridge University Press: 19 August 2009
- Frontmatter
- Contents
- 1 Introduction
- 2 Integrable dynamical systems
- 3 Synopsis of integrable systems
- 4 Algebraic methods
- 5 Analytical methods
- 6 The closed Toda chain
- 7 The Calogero—Moser model
- 8 Isomonodromic deformations
- 9 Grassmannian and integrable hierarchies
- 10 The KP hierarchy
- 11 The KdV hierarchy
- 12 The Toda field theories
- 13 Classical inverse scattering method
- 14 Symplectic geometry
- 15 Riemann surfaces
- 16 Lie algebras
- Index
Summary
We present basic facts about Lie groups and Lie algebras. We describe semi-simple Lie algebras and their representations which can be characterized in terms of roots and weights. We discuss infinite-dimensional Lie algebras, called affine Kac—Moody algebras, which are at the heart of the study of field theoretical integrable systems. In particular we construct the so-called level one representations using the techniques of Fock spaces and vertex operators introduced in Chapter 9.
Lie groups and Lie algebras
A Lie group is a group G which is at the same time a differentiable manifold, and such that the group operation (g, h) → gh-1 is differentiable. Due to a theorem of Montgomery and Zippin, the differentiable structure is automatically real analytic.
The maps h → gh and h → hg are called respectively left and right translations by g. Their differentials at the point h map the tangent space Th(G) respectively to Tgh(G) and Thg(G). We will denote by g · X and X · g the images of X ∊ Th(G) by these maps. This notation is coherent because, differentiating the associativity condition in G, one gets (g · X) · h = g · (X · h), and g · (h · X) = (gh) · X.
- Type
- Chapter
- Information
- Introduction to Classical Integrable Systems , pp. 571 - 598Publisher: Cambridge University PressPrint publication year: 2003