##
**Coxeter graphs and towers of algebras.**
*(English)*
Zbl 0698.46050

Mathematical Sciences Research Institute Publications, 14. New York etc.: Springer-Verlag. vi, 288 p. DM 68.00 (1989).

In his penetrating paper “Index for subfactors” which appeared in Invent. Math. 72, 1-25 (1983; Zbl 0508.46040), V. F. R Jones defined the index \([M_ 1:M_ 0]\), in terms of the Murray-von Neumann coupling constant, for a pair \(M_ 0\subset M_ 1\) of \(II_ 1\) factors and show that the index is either 4 \(cos^ 2(\pi /q)\) for some integer \(q\geq 3\) or it is at least 4.

Apart from laying the groundwork for much subsequent research in von Neumann algebras, it was soon discovered that certain algebras used in this paper could also be used to define a new polynomial invariant in knot theory and further investigations led to important connections with statistical mechanics and quantum field theory.

The subject of the book is certain algebraic and von Neumann algebraic topics closely related to the aforementioned paper of Jones. The central theme is the tower of algebras \[ M_ 0\subset M_ 1\subset...\subset M_ k\subset... \] determined by a pair \(M_ 0\subset M_ 1\) of algebras and obtained by a fundamental construction. The tower has a rich structure and can be used to define various invariants of the pair including the index \([M_ 1:M_ 0]\). The index takes certain values as above and its square root is the norm of an inclusion matrix. Each inclusion matrix can be encoded as a Coxeter graph and those with norm no larger than 2 can be classified, just as the corresponding Coxeter graphs. In the course of the development as well as in the appendices, the contacts with diverse fields of mathematics are explained.

The authors have written a masterly introduction to the subject and their research which cover a wide range of mathematics.

Apart from laying the groundwork for much subsequent research in von Neumann algebras, it was soon discovered that certain algebras used in this paper could also be used to define a new polynomial invariant in knot theory and further investigations led to important connections with statistical mechanics and quantum field theory.

The subject of the book is certain algebraic and von Neumann algebraic topics closely related to the aforementioned paper of Jones. The central theme is the tower of algebras \[ M_ 0\subset M_ 1\subset...\subset M_ k\subset... \] determined by a pair \(M_ 0\subset M_ 1\) of algebras and obtained by a fundamental construction. The tower has a rich structure and can be used to define various invariants of the pair including the index \([M_ 1:M_ 0]\). The index takes certain values as above and its square root is the norm of an inclusion matrix. Each inclusion matrix can be encoded as a Coxeter graph and those with norm no larger than 2 can be classified, just as the corresponding Coxeter graphs. In the course of the development as well as in the appendices, the contacts with diverse fields of mathematics are explained.

The authors have written a masterly introduction to the subject and their research which cover a wide range of mathematics.

Reviewer: C.-h.Chu

### MSC:

46L10 | General theory of von Neumann algebras |

05C50 | Graphs and linear algebra (matrices, eigenvalues, etc.) |

46L35 | Classifications of \(C^*\)-algebras |

46-02 | Research exposition (monographs, survey articles) pertaining to functional analysis |

16Kxx | Division rings and semisimple Artin rings |

81T05 | Axiomatic quantum field theory; operator algebras |