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- Lectures on Infinite Dimensional Lie Algebra;
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New Releases. Description This book is a collection of a series of lectures given by Prof. These lectures focus on the idea of a highest weight representation, which goes through four different incarnations. The second is the highest weight representations of the Lie algebra gl of infinite matrices, along with their applications to the theory of soliton equations, discovered by Sato and Date, Jimbo, Kashiwara and Miwa.

These algebras appear in the lectures twice, in the reduction theory of soliton equations KP KdV and in the Sugawara construction as the main tool in the study of the fourth incarnation of the main idea, the theory of the highest weight representations of the Virasoro algebra. Hereditary operators in Lie algebras are investigated. These are operators which are characterized by a special algebraic equation and their main property is that they generate abelian subalgebras of the given Lie algebra. These abelian subalgebras are infinite dimensional if the hereditary operator is not cyclic.

As a consequence hereditary operators generate on a systematic level nonlinear dynamical systems which possess infinite dimensional abelian groups of symmetry transformations. We show that hereditary operators can be understood as special Lie algebra deformations with a linear interpolation property. In order to construct new hereditary operators out of given ones we study the permanence properties of these operators; this study of permanence properties leads in a natural way to a notion of compatibility.

## ISBN 13: 9789810241292

For local hereditary operators it is shown that eigenvector decompositions are time invariant such an eigenvector decomposition is known to characterize pure multisoliton solutions. Apart from the well-known equations KdV, sine-Gordon, etc. Oxford University Press is a department of the University of Oxford.

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