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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Let M be a multi-matrix algebra over 1<, with minimal central idempotents Pi'· .. 3. Inclusion matrix and Bratteli diagram some integer Pj. , i=1 I and denote by p- or ~ the m-tuple (#1' ... ,p-m)t of dimensions. ) The isomorphism class of M is completely described by the class of ~ modulo permutation of its coordinates. m Observe in particular that the I<-dimension of M, which is is the square of the L,p-r, i=l Euclidean norm of ~. N and ZN with j=1 J ED I

B) CqFq(qMq) = qCF(M)q. Proof. C F (qMq)q. CF(M)q. M)q.. 4. We leave details of (a) to the reader. # Remarks. (1) The algebra qMq is called the reduction of M by q. 5 the hypothesis that q is either in M or in its commutant. 6. Suppose that M is a factor, N and ill' are subfactors of M containing the identity element of M, and cp: N - i ill' is an isomorphism. Then there is an inner automorphism fJ of M such that fJl N = cpo In particular, any automorphism of M is inner. Proof. We identify M with Endl«V) for some vector space V over I<.

48 Chapter 2: Towers of multi-matrix algebras (i) Consider the two subalgebras (both of dimension 62): of the factor M = Mat I2 (K), both inclusions being described by (x,y,z) ... Then [ 00] 0yO. X OOz A~ = AM = (111) though N and 'iii' are not isomorphic. N (ii) Consider N = K (9 Mat 2(K) included in M = Mat 4(K) by (x,y)'" M = Mat 5(K) by (x,y) ... [~ ~ ~]. A~ Then OOy and AM are N (x,y) ... [H HI N = K (9 Mat2(K) and by (x,y) ... OOOy [~ ~ ~]. included and in pseudo-equiVal:~:O (2 1) = Mat 5(K) by but M and M are not isomorphic.

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