##
**Infinite dimensional Lie algebras. An introduction.**
*(English)*
Zbl 0537.17001

Progress in Mathematics, Vol. 44. Boston - Basel - Stuttgart: BirkhĂ¤user. xvi, 245 p. DM 58.00 (1983).

This book is a remarkable contribution to the ”hot” area of Kac-Moody Lie algebras. These algebras appear in many problems of, as Kac points out, mathematical physics, combinatorics, modular forms, etc. Their most spectacular applications are in the areas of Hamiltonian systems and combinatorial identities. Besides being natural and useful they led to the re-examining of many results and proofs in the theory of finite dimensional Lie algebras thereby gaining often new insights into the structure of the latter; this work was done mostly by V. Kac himself, often in collaboration with D. Peterson. The most remarkable are, probably, the recent results of V. Kac on cohomology of compact Lie groups (and related results about Chow rings etc.).

The book concentrates mostly on the areas where V. Kac contributed original work (and these are almost all areas). Some of the areas least covered in the book are applications to PDE, touched upon in the concluding chapter of the book, and cohomology of Kac-Moody Lie algebras, introduced by Garland and Lepowsky.

The book is a very good introduction into the theory. It leads the reader directly through and to main results, ideas, and methods of the theory. In bibliographical notes at the end of each chapter the author gives an additional guidance to the literature. Some results, side-lines and problems are put forward as exercises so that the book does not get bogged down with unnecessary ”completeness” but the reader gets clearly an idea of the state of affairs and receives a thorough orientation.

The Kac-Moody Lie algebras were defined almost simultaneously by V. Kac and R. Moody. They are graded Lie algebras \(G=\oplus_{i\in\mathbb Z}G_i\) (i.e. \([G_i,G_j]\subseteq G_{i+j})\) such that \(G_1\) has basis \(e_1,\ldots,e_n\), \(G_ 0\) has basis \(h_1,\ldots,h_n\), \(G_{-1}\) has basis \(f_ 1,\ldots,f_n\) with (1) \(G_0\) commutative, (2) \([h_i,e_j]=a_{ij}e_j\), \([h_i,f_j]=- a_{ij}f_j\), (3) \([e_i,f_j]=\delta_{ij}h_ i\), (4) \(a_{ij}\) integers, \(a_{ii}=2\), \(a_{ij}\leq 0\) for \(i\neq j\), (5) \(G\) has no graded ideals intersecting nontrivially with \(G_0\).

V. Kac obtained these algebras during his work on classification of infinite-dimensional Lie algebras of Cartan type. At that time the Kac-Moody Lie algebras looked as an interesting but only slightly relevant generalization. The situation changed completely when V. Kac developed, in 1974, his theory of highest weight modules for the Kac-Moody algebras in order to give an independent proof and conceptualization of I. Macdonald’s interpretation of certain combinatorial identities through affine Weyl groups. After this paper the area faced an explosive growth and the Kac-Moody Lie algebras started to appear almost everywhere. V. Kac has, in most directions, led the development of the theory. Now his interests moved to algebraic groups associated to Kac-Moody algebras; a description of results about these groups would take, as V. Kac himself points out, writing another book. His work in this direction was done mostly jointly with D. Peterson.

The Kac-Moody Lie algebras have several important subclasses. One of the smallest (but infinite-dimensional) ones is the class of affine Kac-Moody Lie algebras. These are the algebras which appear most often in the applications. They are related to the Lie algebra of maps of a manifold \(M\) into a finite-dimensional semisimple Lie algebra; one must have \(\dim M=1\). But even that is not enough: \(M\) must be \(S^ 1\). The case of \(\dim M>1\) or \(M\) is a general curve still seem to be intractable.

The next larger class is done when the Cartan matrix \(A=(a_{ij})\) is symmetrizable, i.e. there exists a diagonal non-degenerate \(D\) such that \(DA\) is symmetric. The importance of this class is related to the fact that in this case \(G\) possesses a non-zero invariant bilinear form.

The book starts essentially on the zero level; but throughout the book it is handy to keep in mind many analogues and parallels with other areas of mathematics, most importantly with the theory if finite dimensional simple Lie algebras.

The first several chapters are leisurely (but with accelerating pace) and incisive introduction. In these chapters a stage is set and carefully prepared for subsequent work. In chapters 7 and 8 the author gives realizations of some affine Kac-Moody algebras. The realizations are very important because they permit precise calculations whereas the definitions, given above, through generators and relations would be too unwieldy.

Chapters 9 through 12 form the core of the book. Chapter 9 introduces the most important object in the area: highest weight modules. Among these latter the most important are integrable highest weight modules. To them V. Kac dedicates Chapters 10 and 11. In Chapter 10 he gives a proof of his celebrated character formula which was the origin of much work on Kac-Moody algebras. In Chapter 12 the character formula is made explicit for affine Kac-Moody algebras giving thereby some combinatorial identities. Of course, the character formula gives some ”identities” in non-affine cases as well; but they are not explicit and therefore not identities in the direct meaning of the word.

In Chapter 13 V. Kac describes a connection of character formulas with modular forms and \(\theta\)-functions. In Chapter 14 he gives some applications to KdV-hierarchies.

This is an excellent book which should be read.

The book concentrates mostly on the areas where V. Kac contributed original work (and these are almost all areas). Some of the areas least covered in the book are applications to PDE, touched upon in the concluding chapter of the book, and cohomology of Kac-Moody Lie algebras, introduced by Garland and Lepowsky.

The book is a very good introduction into the theory. It leads the reader directly through and to main results, ideas, and methods of the theory. In bibliographical notes at the end of each chapter the author gives an additional guidance to the literature. Some results, side-lines and problems are put forward as exercises so that the book does not get bogged down with unnecessary ”completeness” but the reader gets clearly an idea of the state of affairs and receives a thorough orientation.

The Kac-Moody Lie algebras were defined almost simultaneously by V. Kac and R. Moody. They are graded Lie algebras \(G=\oplus_{i\in\mathbb Z}G_i\) (i.e. \([G_i,G_j]\subseteq G_{i+j})\) such that \(G_1\) has basis \(e_1,\ldots,e_n\), \(G_ 0\) has basis \(h_1,\ldots,h_n\), \(G_{-1}\) has basis \(f_ 1,\ldots,f_n\) with (1) \(G_0\) commutative, (2) \([h_i,e_j]=a_{ij}e_j\), \([h_i,f_j]=- a_{ij}f_j\), (3) \([e_i,f_j]=\delta_{ij}h_ i\), (4) \(a_{ij}\) integers, \(a_{ii}=2\), \(a_{ij}\leq 0\) for \(i\neq j\), (5) \(G\) has no graded ideals intersecting nontrivially with \(G_0\).

V. Kac obtained these algebras during his work on classification of infinite-dimensional Lie algebras of Cartan type. At that time the Kac-Moody Lie algebras looked as an interesting but only slightly relevant generalization. The situation changed completely when V. Kac developed, in 1974, his theory of highest weight modules for the Kac-Moody algebras in order to give an independent proof and conceptualization of I. Macdonald’s interpretation of certain combinatorial identities through affine Weyl groups. After this paper the area faced an explosive growth and the Kac-Moody Lie algebras started to appear almost everywhere. V. Kac has, in most directions, led the development of the theory. Now his interests moved to algebraic groups associated to Kac-Moody algebras; a description of results about these groups would take, as V. Kac himself points out, writing another book. His work in this direction was done mostly jointly with D. Peterson.

The Kac-Moody Lie algebras have several important subclasses. One of the smallest (but infinite-dimensional) ones is the class of affine Kac-Moody Lie algebras. These are the algebras which appear most often in the applications. They are related to the Lie algebra of maps of a manifold \(M\) into a finite-dimensional semisimple Lie algebra; one must have \(\dim M=1\). But even that is not enough: \(M\) must be \(S^ 1\). The case of \(\dim M>1\) or \(M\) is a general curve still seem to be intractable.

The next larger class is done when the Cartan matrix \(A=(a_{ij})\) is symmetrizable, i.e. there exists a diagonal non-degenerate \(D\) such that \(DA\) is symmetric. The importance of this class is related to the fact that in this case \(G\) possesses a non-zero invariant bilinear form.

The book starts essentially on the zero level; but throughout the book it is handy to keep in mind many analogues and parallels with other areas of mathematics, most importantly with the theory if finite dimensional simple Lie algebras.

The first several chapters are leisurely (but with accelerating pace) and incisive introduction. In these chapters a stage is set and carefully prepared for subsequent work. In chapters 7 and 8 the author gives realizations of some affine Kac-Moody algebras. The realizations are very important because they permit precise calculations whereas the definitions, given above, through generators and relations would be too unwieldy.

Chapters 9 through 12 form the core of the book. Chapter 9 introduces the most important object in the area: highest weight modules. Among these latter the most important are integrable highest weight modules. To them V. Kac dedicates Chapters 10 and 11. In Chapter 10 he gives a proof of his celebrated character formula which was the origin of much work on Kac-Moody algebras. In Chapter 12 the character formula is made explicit for affine Kac-Moody algebras giving thereby some combinatorial identities. Of course, the character formula gives some ”identities” in non-affine cases as well; but they are not explicit and therefore not identities in the direct meaning of the word.

In Chapter 13 V. Kac describes a connection of character formulas with modular forms and \(\theta\)-functions. In Chapter 14 he gives some applications to KdV-hierarchies.

This is an excellent book which should be read.

Reviewer: B. Ju. Weisfeiler

### MSC:

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

17-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to nonassociative rings and algebras |

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

22E67 | Loop groups and related constructions, group-theoretic treatment |

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

22-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to topological groups |

05A19 | Combinatorial identities, bijective combinatorics |

11P81 | Elementary theory of partitions |

11F22 | Relationship to Lie algebras and finite simple groups |

35Q53 | KdV equations (Korteweg-de Vries equations) |

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 |