Current Algebras and Groups

1. Front Matter

   Pages i-xvii

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2. Semisimple Lie Algebras

   • Jouko Mickelsson

   Pages 1-20

3. Representations of Affine Kac-Moody Algebras

   • Jouko Mickelsson

   Pages 21-42

4. Principal Bundles

   • Jouko Mickelsson

   Pages 43-74

5. Extensions of Groups of Gauge Transformations

   • Jouko Mickelsson

   Pages 75-104

6. The Chiral Anomaly

   • Jouko Mickelsson

   Pages 105-126

7. Determinant Bundles over Grassmannians

   • Jouko Mickelsson

   Pages 127-170

8. The Virasoro Algebra

   • Jouko Mickelsson

   Pages 171-191

9. The Boson Fermion Correspondence

   • Jouko Mickelsson

   Pages 193-202

10. Holomorphic Aspects of String Theory

    • Jouko Mickelsson

    Pages 203-233

11. The Nonlinear σ Model

    • Jouko Mickelsson

    Pages 235-244

12. The Kp Hierarchy

    • Jouko Mickelsson

    Pages 245-265

13. The Fock Bundle of a Dirac Operator and Infinite Grassmannians

    • Jouko Mickelsson

    Pages 267-297

14. Back Matter

    Pages 299-313

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Let M be a smooth manifold and G a Lie group. In this book we shall study infinite-dimensional Lie algebras associated both to the group Map(M, G) of smooth mappings from M to G and to the group of dif­ feomorphisms of M. In the former case the Lie algebra of the group is the algebra Mg of smooth mappings from M to the Lie algebra gof G. In the latter case the Lie algebra is the algebra Vect M of smooth vector fields on M. However, it turns out that in many applications to field theory and statistical physics one must deal with certain extensions of the above mentioned Lie algebras. In the simplest case M is the unit circle SI, G is a simple finite­ dimensional Lie group and the central extension of Map( SI, g) is an affine Kac-Moody algebra. The highest weight theory of finite­ dimensional Lie algebras can be extended to the case of an affine Lie algebra. The important point is that Map(Sl, g) can be split to positive and negative Fourier modes and the finite-dimensional piece g corre­ sponding to the zero mode.

• algebra
• model
• operator
• statistical physics
• transformation

• University of Jyväskylä, Jyväskylä, Finland

  Jouko Mickelsson

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