# Coxeter Graphs and Towers of Algebras by F. M. Goodman, P. de la Harpe, V. F. R. Jones

By F. M. Goodman, P. de la Harpe, V. F. R. Jones

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Thus v,J ® ei,i = y ® 1 for some y E Endl«W), Since in addition x commutes with i A ® I, it follows that yEA'. On the other hand any element of the form y ® 1, yEA' , commutes with A ~ Matd(I<). # Chapter 2: Towers of multi-matrix algebras 38 Consider a factor F ~ Endl«V), where V is a finite dimensional I<-vector space. The basic facts about the representation theory of Fare: (i) Any finite dimensional F-module is completely reducible, and (ii) any irreducible F-module is eqnivalent to the standard module V.

Xn) is diag(xj, ... ,xj), with Xj being repeated \ j times; cf. 2. Hence there is a basi s of Vi such that the matrix of 'lfi (xl' ... ,xn ) is with xJ. being repeated A.. times. I,J Next we introduce some terminology in order to give the K-theoretic interpretation of the inclusion matrix. The KO group of a ring R is an abelian group constructed from the representation theory of R; the set of equivalence classes of finitely generated projective R-modules forms an abelian semigroup, the operation being direct sum, and KO(R) is the quotient group of this semigroup.

F) If dimK(ZM n ZN) ) 1, there exists an idempotent r E ZM r of 1. With an appropriate indexing of the Pi's and qj' s, one has n ZN with r"l 0 and Chapter 2: Towers of multi-matrix algebras 46 PI'·· ·,Pm,jql'·· ·,qn' E rMr Pm'+l'···'Pm jqn'+1,···,qn E (I-r)M(I-i:) for some m' ,n' with 0 < m' < m and 0 < n' < n. It follows that Mi,j = 0 and '\j = 0 unless either I ~ i ~ m' and I ~ j s n' ill: m' + I ~ i ~ m and n' +1 s j s n so that A~ is decomposable. One checks conversely that, if A~ is decomposable, then ZM n ZN contains a nontrivial idempotent which is a sum of some Pi's and also a sum of some qj's.