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Editorial Reviews
This book is a collection of a series of lectures given by Prof. V Kac at Tata Institute, India in Dec '85 and Jan '86. These lectures focus on the idea of a highest weight representation, which goes through four different incarnations.
The first is the canonical commutation relations of the infinite-dimensional Heisenberg Algebra (= ocillator algebra). 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. The third is the unitary highest weight representations of the current (= affine Kac-Moody) algebras. 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.
This book should be very useful for both mathematicians and physicists. To mathematicians, it illustrates the interaction of the key ideas of the representation theory of infinite-dimensional Lie algebras; and to physicists, this theory is turning into an important component of such domains of theoretical physics as soliton theory, theory of two-dimensional statistical models, and string theory. --This text refers to the Paperback edition.
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We shall first consider the Virasoro algebra, which is playing an increasingly important role in theoretical physics. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
unitary highest weight representation, highest weight representations, principal gradation, oscillator algebra, contravariant form, vacuum vector, highest weight vector, determinant formula, highest component, fundamental weights, dimensional lie algebras, character formula, singular vector, unitary representations, central extension, central charge, commutation relations, normal ordering Key Phrases - Capitalized Phrases (CAPs): (learn more)
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