Titlebook: Infinite Dimensional Lie Algebras; An Introduction Victor G. Kac Book 1983 Springer Science+Business Media New York 1983 cls.differential e

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https://doi.org/10.1007/978-1-4757-1382-4cls; differential equation; invariant; partial differential equation; combinatorics

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Affine Lie algebras: the realization (case k=2 or 3). Application to the classification of finite o polynomial maps from it ℂ. to a simple finite-dimensional Lie algebra with the action of a finite cyclic group. As a side result of this construction we deduce a nice description of the finite order automorphisms of a simple finite-dimensional Lie algebra, and, in particular, the classification of symmetric spaces.

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The principal realization of the basic representation. Application to the KdV-type hierarchies of ntly in terms of certain (infinite order) differential operators in infinitely many indeterminates, called the vertex operators. The so-called principal Heisenberg subalgebra s of g(.) plays a crucial role in this construction. In a similar fashion, we construct representations of affine Lie algebras of infinite rank.

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Real and imaginary roots,In this chapter we give an explicit description of the root system ∆ of a Kac-Moody algebra g(A). Our main instrument is the notion of an imaginary root, which has no counterpart in the finite-dimensional theory.

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Integrable highest weight modules over affine Lie algebras. Application to ,-function identities,In the last three chapters we developed a representation theory of arbitrary Kac-Moody algebras. From now on we turn to the special case of affine Lie algebras.

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