## Bombay Lectures on Highest Weight Representations of Infinite Dimensional Lie AlgebrasThis 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 (= oscillator 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. |

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### Contents

Section 1 | 1 |

Section 2 | 11 |

Section 3 | 19 |

Section 4 | 33 |

Section 5 | 49 |

Section 6 | 59 |

Section 7 | 63 |

Section 8 | 69 |

Section 9 | 72 |

Section 10 | 75 |

Section 11 | 81 |

Section 12 | 105 |

Section 13 | 117 |

Section 14 | 129 |

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### Common terms and phrases

adjoint affine algebra algebra gin antilinear antilinear anti-involution basis bilinear form central charge central extension coefficient commutation relations completes the proof construction contravariant contravariant Hermitian form Corollary decomposition defined Definition denote diagonal dimensional direct sum discrete series eigenspace of L0 electron elements finite number finite-dimensional follows function given Hence Hermitian form highest component highest weight representation highest weight vector infinite invariant irreducible highest weight irreducible representation isomorphism Kac determinant formula KP hierarchy Lemma Lie algebra linear span linearly independent loop algebra maps matrix monomials multiple nonzero normal ordering Note obtain oscillator algebra oscillator representation proper subrepresentation proved Recall region Remark repre representation of Vir Res0 right-hand side satisfy scalar Schur polynomials semi-infinite monomials sentation singular vector solution subspace tensor product Theorem tion unitary highest weight unitary representation vacuum vector Vect vector space Virasoro algebra