» Kac V. J., Raina A. K. - Highest-weight representations of infinite-dimensional Lie algebras
Kac V. J., Raina A. K. - Highest-weight representations of infinite-dimensional Lie algebras
||Highest-weight representations of infinite-dimensional Lie algebras
||Kac V. J., Raina A. K.
||This book is a write-up of a series of lectures given by the Hist attthoi ut Tata Institute, Bombay, during December 1985 - Jaimaiy 1986.
The dominant tlteme of these lecttues is the idea of a highest weight icj scntatioti. This idea goes through four different incarnations.
The first is the canonical commutation relations of the infinite-dimensiu Heiscnberg algebra (= oscillator algebra). Although this example ь exiienii simple, it not only contains the germs of the main featuies of the tlieoiy, I also serves as a basis for most of the constructions of representations ol lnlini dimensional Lie algebras.
The second is the highest weight representations of the Lie algebra^H^ of finite matrices, along with their applications to the theory of soliton equatioi discovered by Sato and Date-Jimbo-Kashiwara-Miwa. Here the main point the isomorphism between the vertex and the "Dirac sea" realizations of t fundamental representations ofgfl^ , a kind of a Bose-Fermi coiiespondence.
The third is the unitary highest weight representations of the affme Ki Moody (= current) algebras. Since there is now a book devoted to the theo of Kac-Moody algebras, it was decided to devote to them a minimum atieutio In the lectures affine algebras play a prominent role only in the Stigawaia to struction as the main tool in the study of the fourth incarnation of the ma idea, the theory of highest weight representations of the Virasoio algebra.
The main results of the representation theory of the Virasoro algebra whk are proved in these lectures are die Kac determinant formula and the uniiaul of the "discrete series" representations of Belavin-Polyakov-Zaniolodchikov an Fried an-Qiu-Shenker.
We hope that this elementary introduction to the subject, written by mathematician and a physicist, will prove useful to both mathematicians an physicists. To mathematicians, since it illustrates, on important examples, ill interaction of the key ideas of the representation theory of infinite dimension. Lie algebras; and to physicists, since this theory is turning before om veiy eye