##
**Infinite dimensional Lie algebras.
3rd ed.**
*(English)*
Zbl 0716.17022

Cambridge etc.: Cambridge University Press. xxi, 400 p. £40.00; $ 69.50 (1990).

[For a review of the first ed. 1983, see Zbl 0537.17001; the second edition was published in 1985, see Zbl 0574.17010.]

From the preface to the third edition: “This edition differs considerably from the previous ones. Particularly, more emphasis is made on connections to mathematical physics, especially to conformal field theory.” There are changes and additions in almost all chapters, the most important in the last three.

In chapter 12 the Sugawara and the coset construction as the basic constructions of conformal field theory are added. Branching functions are introduced. In chapter 13 a study of general branching functions along with string functions is added. There are estimates of the order of poles and of levels of the branching functions. Modular and conformal invariance is used to study unitarizable representations of the Virasoro algebra. Elements of the representation theory of the Virasoro are already introduced in chapter 9 together with a free field construction of representations of the Virasoro algebra. Unitarizability considerations of representations of the Virasoro algebra are added in chapter 11. In chapter 14 the vertex operator calculus is used for the homogeneous vertex operator construction in the untwisted case. The basic representation of the infinite rank affine algebra of type \(A_{\infty}\) is constructed via the so called boson-fermion correspondence. This leads to a discussion of the KP-hierarchy. The soliton equations of the KdV and NLS hierarchies are described with the help of the principal and homogeneous vertex operator constructions of \(A_1^{(1)}.\)

There are a lot of new exercises. As an example of quantized Kac-Moody algebras a discussion of the algebra \(U_q(sl_2)\) in several exercises is added in chapter 3. A sketchy theory of infinite Grassmannian and flag manifolds and its connections to KP and MKP hierarchies is given in exercises in chapter 14. There is a basis free theory of the Lie algebra and group of type \(A_{\infty}\) in exercises using results of Kac-Peterson and Pressley-Segal. There are also additional exercises without connection to mathematical physics. As an application classical theorems of the theory of algebraic curves of Chevalley and Tate are derived.

The list of references and bibliographical notes is updated.

The third edition is an important enlargement of the valuable standard book.

From the preface to the third edition: “This edition differs considerably from the previous ones. Particularly, more emphasis is made on connections to mathematical physics, especially to conformal field theory.” There are changes and additions in almost all chapters, the most important in the last three.

In chapter 12 the Sugawara and the coset construction as the basic constructions of conformal field theory are added. Branching functions are introduced. In chapter 13 a study of general branching functions along with string functions is added. There are estimates of the order of poles and of levels of the branching functions. Modular and conformal invariance is used to study unitarizable representations of the Virasoro algebra. Elements of the representation theory of the Virasoro are already introduced in chapter 9 together with a free field construction of representations of the Virasoro algebra. Unitarizability considerations of representations of the Virasoro algebra are added in chapter 11. In chapter 14 the vertex operator calculus is used for the homogeneous vertex operator construction in the untwisted case. The basic representation of the infinite rank affine algebra of type \(A_{\infty}\) is constructed via the so called boson-fermion correspondence. This leads to a discussion of the KP-hierarchy. The soliton equations of the KdV and NLS hierarchies are described with the help of the principal and homogeneous vertex operator constructions of \(A_1^{(1)}.\)

There are a lot of new exercises. As an example of quantized Kac-Moody algebras a discussion of the algebra \(U_q(sl_2)\) in several exercises is added in chapter 3. A sketchy theory of infinite Grassmannian and flag manifolds and its connections to KP and MKP hierarchies is given in exercises in chapter 14. There is a basis free theory of the Lie algebra and group of type \(A_{\infty}\) in exercises using results of Kac-Peterson and Pressley-Segal. There are also additional exercises without connection to mathematical physics. As an application classical theorems of the theory of algebraic curves of Chevalley and Tate are derived.

The list of references and bibliographical notes is updated.

The third edition is an important enlargement of the valuable standard book.

Reviewer: Helmut Boseck (Greifswald)

### MSC:

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

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

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

17B68 | Virasoro and related algebras |

17B69 | Vertex operators; vertex operator algebras and related structures |

81R10 | Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations |