##
**Bombay lectures on highest weight representations of infinite dimensional Lie algebras.**
*(English)*
Zbl 0668.17012

Advanced Series in Mathematical Physics, Vol. 2. Singapore-New Jersey-Hong Kong: World Scientific, ix, 145 p. (1987).

The main topic of this book is the theory of highest weight representations of the Virasoro algebra. Three methods of constructing such representations are discussed:

(1) in terms of “normal ordered” products of operators in the canonical representation of an infinite-dimensional Heisenberg Lie algebra;

(2) by embedding the Virasoro algebra in the universal central extension of \(\mathfrak{gl}_{\infty}\), the Lie algebra of an infinite general linear group, and restricting highest weight representations of \(\mathfrak{gl}_{\infty}\) to the Virasoro algebra;

(3) the so-called Sugawara construction, and its generalization due to Goddard, Kent and Olive, which provides a representation of the Virasoro algebra associated to any integrable highest weight representation of an affine Lie algebra, possibly together with its restriction to an affine Lie subalgebra.

Particular emphasis is placed on the question of which highest weight representations of the Virasoro algebra are unitary. The authors do not give a complete proof of the result of Friedan, Qiu and Shenker which gives a necessary condition for unitarity, but the condition is proved to be sufficient. This involves a discussion of the Kac determinant formula, which gives the determinant of the unique invariant Hermitian form carried by any highest weight representation of the Virasoro algebra. A complete proof of this formula is given.

Other topics discussed include the boson-fermion correspondence and the relation between representations of \(\mathfrak{gl}_{\infty}\) and soliton equations. There is no index.

(1) in terms of “normal ordered” products of operators in the canonical representation of an infinite-dimensional Heisenberg Lie algebra;

(2) by embedding the Virasoro algebra in the universal central extension of \(\mathfrak{gl}_{\infty}\), the Lie algebra of an infinite general linear group, and restricting highest weight representations of \(\mathfrak{gl}_{\infty}\) to the Virasoro algebra;

(3) the so-called Sugawara construction, and its generalization due to Goddard, Kent and Olive, which provides a representation of the Virasoro algebra associated to any integrable highest weight representation of an affine Lie algebra, possibly together with its restriction to an affine Lie subalgebra.

Particular emphasis is placed on the question of which highest weight representations of the Virasoro algebra are unitary. The authors do not give a complete proof of the result of Friedan, Qiu and Shenker which gives a necessary condition for unitarity, but the condition is proved to be sufficient. This involves a discussion of the Kac determinant formula, which gives the determinant of the unique invariant Hermitian form carried by any highest weight representation of the Virasoro algebra. A complete proof of this formula is given.

Other topics discussed include the boson-fermion correspondence and the relation between representations of \(\mathfrak{gl}_{\infty}\) and soliton equations. There is no index.

Reviewer: Shrawan Kumar (Bombay)

### MSC:

17B65 | Infinite-dimensional Lie (super)algebras |

17B67 | Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras |

17B68 | Virasoro and related algebras |

37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |

37K30 | Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with infinite-dimensional Lie algebras and other algebraic structures |

37K40 | Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems |

17-02 | Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras |

17B10 | Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) |