Loop groups.

*(English)*Zbl 0618.22011
Oxford Mathematical Monographs. Oxford: Clarendon Press. VIII, 318 p. £40.00 (1986).

A loop group LG is the group of maps of the circle \(S^ 1\) into some topological group G (with group law coming from pointwise multiplication in G). The book under review is devoted to the study of structure and representations of LG in the case when G is either a compact or a complex Lie group.

Groups similar to LG enter mathematics in several points. Their Lie algebras form a class of Kac-Moody Lie algebras, the so-called affine algebras. In the last few years these algebras were extensively studied, and their deep relations with various branches of mathematics (combinatorics, finite groups) and physics (quantum field theory, especially string models) were displayed [see, for example, Vertex operators in mathematics and physics, J. Lepowsky, S. Mandelstam and I. M. Singer (eds.) (Publ., Math. Sci. Res. Inst. 3) (1985; Zbl 0549.00013)].

The present book differs from the other sources in that it mostly adopts analytic and geometric, rather that algebraic and combinatoric, approaches.

The first part of the book studies the group LG itself. After the introduction (Chapter I) and a survey of the results about finite- dimensional representations of Lie groups (Chapter 2) the authors give in Chapter 3 general facts about infinite-dimensional Lie groups and consider LG from this viewpoint. In Chapter 4 they study one of the most important properites of loop groups, namely the existence of natural central extensions by the circle group T; these extensions are sometimes more important than loop groups themselves. In this chapter the extensions are constructed and studied by differential geometric methods. Chapter 5 contains a brief survey of Kac-Moody Lie algebras as Lie algebras of loop groups.

Chapter 6 is one of the main chapters in the first part of the book. In this chapter LG is represented as a group of operators in an appropriate Hilbert space, namely in the space \(H=L^ 2(S^ 1,V)\) of \(L^ 2\)- functions on the circle with values in some finite-dimensional representation of G; LG acts in H pointwise. The idea (coming from quantum field theory) is to consider the polarization of H, i.e. the decomposition \(H=H_+\oplus H_-\) where \(H_+\) (resp. \(H_-)\) is the space of functions with vanishing negative (resp. positive) Fourier coefficients. Properties of operators from LG with respect to this decomposition form a very interesting and important part of the theory.

Another very important concept in the first part of the book is the notion of the Grassmannian Gr(H) of a polarized Hilbert space H, introduced in Chapter 7. The authors study the canonical (determinant) line bundle on Gr(H), Schubert cell decomposition of Gr(H), etc.

Chapter 8 introduces the fundamental homogeneous space X of LG that is defined as LG/G where \(G\subset LG\) is considered as the subgroup of constant loops. Two main properties of X are as follows. First, X can be considered as a (infinite-dimensional) complex manifold via the identification \(X=LG_{{\mathbb{C}}}/L_+G_{{\mathbb{C}}}\) where \(G_{{\mathbb{C}}}\) is the complexification of G and \(L_+G_{{\mathbb{C}}}\) consists of boundary values of analytic mappings of the disk \(| z| <1\) into \(G_{{\mathbb{C}}}\). Second, X can be canonically imbedded into Gr(H), thus inheriting from Gr(H) the stratification by Schubert cells and other nice features.

The remaining Chapters 9-14 deal with the representation theory of loop groups. We will not describe the content of these chapters and restrict ourselves to giving their titles: Ch. 9: Representation theory. Ch. 10: The fundamental representation. Ch. 11: The Borel-Weil theory. Ch. 12: The spin representation. Ch. 13: ”Blips” or ”vertex” operators. Ch. 14: The Kac character formula and the Bernstein-Gel’fand-Gel’fand resolution.

Groups similar to LG enter mathematics in several points. Their Lie algebras form a class of Kac-Moody Lie algebras, the so-called affine algebras. In the last few years these algebras were extensively studied, and their deep relations with various branches of mathematics (combinatorics, finite groups) and physics (quantum field theory, especially string models) were displayed [see, for example, Vertex operators in mathematics and physics, J. Lepowsky, S. Mandelstam and I. M. Singer (eds.) (Publ., Math. Sci. Res. Inst. 3) (1985; Zbl 0549.00013)].

The present book differs from the other sources in that it mostly adopts analytic and geometric, rather that algebraic and combinatoric, approaches.

The first part of the book studies the group LG itself. After the introduction (Chapter I) and a survey of the results about finite- dimensional representations of Lie groups (Chapter 2) the authors give in Chapter 3 general facts about infinite-dimensional Lie groups and consider LG from this viewpoint. In Chapter 4 they study one of the most important properites of loop groups, namely the existence of natural central extensions by the circle group T; these extensions are sometimes more important than loop groups themselves. In this chapter the extensions are constructed and studied by differential geometric methods. Chapter 5 contains a brief survey of Kac-Moody Lie algebras as Lie algebras of loop groups.

Chapter 6 is one of the main chapters in the first part of the book. In this chapter LG is represented as a group of operators in an appropriate Hilbert space, namely in the space \(H=L^ 2(S^ 1,V)\) of \(L^ 2\)- functions on the circle with values in some finite-dimensional representation of G; LG acts in H pointwise. The idea (coming from quantum field theory) is to consider the polarization of H, i.e. the decomposition \(H=H_+\oplus H_-\) where \(H_+\) (resp. \(H_-)\) is the space of functions with vanishing negative (resp. positive) Fourier coefficients. Properties of operators from LG with respect to this decomposition form a very interesting and important part of the theory.

Another very important concept in the first part of the book is the notion of the Grassmannian Gr(H) of a polarized Hilbert space H, introduced in Chapter 7. The authors study the canonical (determinant) line bundle on Gr(H), Schubert cell decomposition of Gr(H), etc.

Chapter 8 introduces the fundamental homogeneous space X of LG that is defined as LG/G where \(G\subset LG\) is considered as the subgroup of constant loops. Two main properties of X are as follows. First, X can be considered as a (infinite-dimensional) complex manifold via the identification \(X=LG_{{\mathbb{C}}}/L_+G_{{\mathbb{C}}}\) where \(G_{{\mathbb{C}}}\) is the complexification of G and \(L_+G_{{\mathbb{C}}}\) consists of boundary values of analytic mappings of the disk \(| z| <1\) into \(G_{{\mathbb{C}}}\). Second, X can be canonically imbedded into Gr(H), thus inheriting from Gr(H) the stratification by Schubert cells and other nice features.

The remaining Chapters 9-14 deal with the representation theory of loop groups. We will not describe the content of these chapters and restrict ourselves to giving their titles: Ch. 9: Representation theory. Ch. 10: The fundamental representation. Ch. 11: The Borel-Weil theory. Ch. 12: The spin representation. Ch. 13: ”Blips” or ”vertex” operators. Ch. 14: The Kac character formula and the Bernstein-Gel’fand-Gel’fand resolution.

Reviewer: S.I.Gel’fand

##### MSC:

22E65 | Infinite-dimensional Lie groups and their Lie algebras: general properties |

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

22-02 | Research exposition (monographs, survey articles) pertaining to topological groups |

46M20 | Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.) |

14M15 | Grassmannians, Schubert varieties, flag manifolds |

43A85 | Harmonic analysis on homogeneous spaces |

32M10 | Homogeneous complex manifolds |