Coxeter Graphs and Towers of Algebras, Goodman, De La Harpe and Jones

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Coxeter Graphs and Towers of AlgebrasL

Frederick M. Goodman Pierre de la Harpe Vaughan F. R. Jones

Springer-Verlag New York Berlin Heidelberg London Paris Tokyo

i,. .Frederick M. Goodman Department of Mathematics University of Iowa Iowa City, Iowa 52240 USA Vaughan F.R. Jones Department d Mathematics University of California - Berkeley 94720 Berkeley, Cal~fomia Pierre de la Harpe Section de Mathimatiques Universiti de Genkve CH- 1211 Genkve 24 Switzerland

PREFACEThe paper [Jol] on subfactors of von Neumam factors has stimulated much research in von Neumann algebras. Quite unexpectedly, it was discovered soon after the appearance of [Jol] that certain algebras which are used there for the analysis of subfactors could also be used to define a new polynomial invariant for links [Jo~].The period of activity following this discovery saw the creation of a number of related invariants as well as the successful use of these invariants in knot theory. Furthermore, rece11t effort to understand the fundamental nature of the the new link invariants has l& to connections with invariant theory, statistical mechanics and quantum field theory. In turn the link invariants, the notion of a quantum group, and the quantum Yang-Baxter equation have had great impact on the study of subfactors. account of these developments, and we It is not yet the time to give a comprehe~sive make no attempt to do so here. Our subject is certain algebraic and von Neurnam algebraic topics closely related to the original paper [Jol]. Hbwever, in order to promote, in a modest way, the contact between diverse fields of mathematics, we have tried to make this work accessible to the broadest audience. Consequently, this book contains much elementary expositoG material. We give here a brief preview of the book. Each of the four chapters has its own introduction, with a more thorough description of the contents. Chapter 1 begins with a (slightly new guise of) a familiar combinatorial problem: to classify finite matrices over the non-negative integers which have Euclidean norm no greater than 2. These are classified by the ubiquitous Cozeter graphs of type A, D, or E (see [HHSV] other occurrences of these graphs) and the set of possible nomu is for (2) u {2ws7r/q : q 2 2). The central theme of the book - the discussion of which begins in Chapter 2 -is the tower of algebras MOc M1 c - ,c Mk c determined by a pair MO C Ml of algebras

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Mathematical Subject Classification (1980): 46L10, DSC50, 16A40 Library of Congress Cataloging-in-Publication Data Goodman, Frederick M. Coxeter graphs and towers of algebras. (MathematicaI,Sciences Research Institute publications ; 14) Bibliography: p. 1. Class field towers. 2. Coxeter graphs. I. La Harpe, Pierre de. 11. Jones, Vaughan F.R., 1952. III. Title. IV. Series. QA247.G68 1989 512'.55 89-5991 Printed on acid-free paper.O 1989by Springer-Verlag New York Inc. All rights reserved. This work may not be translated or copied in whde or in part withoutthe written permission of the publisher (Springer-Verlag, 175 Fifth Avenue, New York, bJY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information-storage and retrieval, electronic adaptation, computer software, or b y , W a r or dissimilar metliodol6gy now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc. in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used M y by anyone.





(with the same identity). The tower can be used to define various invariants of the pair, including the index [M1:MO] In Chapters 2 and 3, we study two cases of the tower construction in detail. In Chapter 2, the algebras are finite direct sums of full matrix algebras over some field. A pair Mo c M1 is described, up to isomorphism, by an inclusion matrix A with non-negative integer entries. This matrix may be encoded as a graph, known as the Bratteli diagram of the pair. It turns out that the index [M1:Md equals 1 1 ~ thus it ~ ~ ; follows from Chapter 1 that [M1:Md 5 4 if and only if the Bratteli diagram is a Coxeter graph of type A, D, or E.



Camera-ready copy prepared by the authors. Printed and bound by R. R. DonneUey & Sons, Harrisonburg, Virginia. Printed in the United States of America. 987654321 ISBN 0-387-96979-9 Springer-Yerlag New York Berlin Heidelberg ISBN 3-540-%979-9 Springer-Verlag Berlin Heidelberg New York





In Chapter 3, the algebras are finite von Neumann algebras with finite dimensional centers. Somewhat surprisingly, the results of Chapter 2 essentially extend to this setting. But now a pair Eho c M1 is (partially) described by an inclusion matrix A with entries in {2cosn/q : q 2 2) U {I : r 2 21, and pairs with index no greater than 4 are associated to Coxeter graphs of arbitrary type, including types B,F,G,H,I. Fnly Chapter 4 is a M h e r analysis of pairs N c M of finite factors of finite index. ial, There are two main themes. The first is the notion of a commuting square, due to Popa [Popl], and its use in approximating pairs of hyperfinite 1 1 factors simultaneously by 1 finite dimensional von Neumann algebras. The second theme is the derived t o w s of a pair of II, factors, which is the c h i n of (necessarily finite dimensional) rdative cornmutants Mo' 17Mk in the tower. All the information in the derived tower can be encoded in a (possibly infinite) graph, the principal graph of the pair. When the index is less than 4, the graph is a Coxeter graph of type A, D, or E. In Chapter 2, we also describe how a certain quotients of the Hecke algebra of type A appear in the tower construction associated to a pair Mo c M1 of, say, finite dimensional semi-simple algebras over the field C. For each choice of a positive faithful trace tr on MI, there is a unique trace preserving conditional expectation El : M1 --,MO, and it t w m out that Ma is naturally generated by M1 and El. Now if the trace tr satisfies

There are several appendices. Appendix I extends the cornputations of Chapter 1. Appendix ILa relates complex semi-~imple algebras and finite dimensional c*-algebras: Appendix X b explains one appearance of the algebras AD,k in statistid mw -. Appendix 1I.c is a further discussion of


for special values of jl.

Appendix 1 1is an exposition of Hecke subgroups in PSL2(R), and thus another famous 1 occurrence of the sequence (2cos(~/k))~,~.


It is a pleasure to record our gratitude to numerous fiends and colleagues for their generous help, including: , R. Baldi, D. Bichsel, H. Dherete, M. Kewaire, A. Ocneanu, M. Pimsner, S. Popa, G. Skandalis, C. Skau, R. Steinberg, V. Sunder, A. Valette, and H. Wenzl. We-gatefully acknowledgesupport from the MSRI in Berkeley, the IHES in Bures, the United States NSF, ttie IMA in Minneapolis, and our home institutions during our work on this project.


the so-called Markov condition, then the situation propagates up the tower, and each is naturally generated by Mk and a conditional expectation algebra Mk+l

Ek : Mk --IMk-l, for aU k 2 1. Moreover the Ekl s are idempotents which satisfy the"braiding" relations pEiEjEi=Ei EiEj = EjEi where {l,E1,.9

ifli-j(=l,and if J i - j J 22,

0 = [M1:MO].Ek-l)

The abstract algebra


presented by generators

and relations as above is a quotient of the Hecke algebra Hk(d), where

q E E satisfies

p ='"2 q + q-l. Although we do not discuss this in the text, we might remark here that the milp




Hm(q)4 alg {l,El,.

.) A c

where Em is the inductive limit of the braid groups Bk, is, up to a normalization, the Jones link invariant [Jo~]. Also let us point out that to obtain the Jones invariant in this way, it is necessary to deal only with finite dimensional algebras, not the less familiar intinite dimensional von Neumann factors._--,

CONTENTSPreface 1, then s+s-l, are in the limit set of E. (iii) The smallest limit point of E is 2 = l i m 2 cos(z); there ezists d > 2 witht

with lull < l,...,lvel

s 1, and consequently with

/' 3

1 -

As the aj' s are in 2, the claim is clear. P(T) = P1(T) = n y - p j ) mi, for any k iL, set = nL~-pi). 1sJS 1r J L Then Pk E H[T] by Newton's formulas for symmetric polynomials (see, e.g., [BA4], page %ti A.IV.57). It follows from the begiining of the proof that P . = Pk for some j,k withJ 1 r j < k, and thus that there exists a prmutation a ,of {1;2,,. .iZ) with



E n ]2,d[ = 4.(iv) The smallest limit point of limit points of E ispoints of E in the interval ]2,AJ

a =? b


2,058171. Limit

c o ~ t m t b increasing injnite sequence an

furthermore for each q 2 3, there are increasing sequences and decreasing sequences in E converging towards X(' 1


(v) The closure of E contains (vi) E is not closed.

[X,O ,m [ . .

Statement (i) is proved in Section 1.5. See also [HW] for other operations defined inside E. statements (ii) to (iv) are proved in Appendix 1. Statement (v) is a recent result of J. Shearer [She]. For (vi), see Appendix I, after 1.3.6. What is known of E . makes it look somehow similar to another set recently studied by [Smy]; see also Lehmer'e problem, referred to below in remark (3) following Proposition 1.3.4, in Appendix I. 1.2. Proof of Kronecker'a themem. We repeat firat the two