Coxeter Graphs and Towers of Algebras

1. Matrices over the natural numbers: values of the norm, classification, and variations.- 1.1. Introduction.- 1.2. Proof of Kronecker's theorem.- 1.3. Decomposability and pseudo-equivalence.- 1.4. Graphs with norms no larger than 2.- 1.5. The set E of norms of graphs and integral matrices.- 2. Towers of multi-matrix algebras.- 2.1. Introduction.- 2.2. Commutant and bicommutant.- 2.3. Inclusion matrix and Bratteli diagram for inclusions of multi-matrix algebras.- 2.4. The fundamental construction and towers for multi-matrix algebras.- 2.5. Traces.- 2.6. Conditional expectations.- 2.7. Markov traces on pairs of multi-matrix algebras.- 2.8. The algebras A?,k for generic ?.- 2.9. An approach to the non-generic case.- 2.10. A digression on Hecke algebras.- 2.10.a. The complex Hecke algebra defined by GLn(q) and its Borel subgroup.- 2.10.b. The Hecke algebras Hq,n.- 2.10.c. Complex representations of the symmetric group.- 2.10.d. Irreducible representations of Hq,n for q ? ?.- 2.11. The relationship between A?,n and the Hecke algebras.- 3. Finite von Neumann algebras with finite dimensional centers.- 3.1. Introduction.- 3.2. The coupling constant: definition.- 3.3. The coupling constant: examples.- 3.3.a. Discrete series.- 3.3.b. Factors defined by icc groups.- 3.3.c. W*(?)-modules associated to subrepresentations of ?G.- 3.3.d. The formula dim?(H) = covol(?) d?.- 3.3.e. A digression on the Peterson inner product.- 3.4. Index for subfactors of II1 factors.- 3.5. Inclusions of finite von Neumann algebras with finite dimensional centers.- 3.6. The fundamental construction.- 3.7. Markov traces on EndN(M), a generalization of index.- 4. Commuting squares, subfactors, and the derived tower.- 4.1. Introduction.- 4.2. Commuting squares.- 4.3. Wenzl's index formula.- 4.4. Examples of irreducible pairs of factors of index less than 4, and a lemma of C. Skau.- 4.5. More examples of irreducible paris of factors, and the index value 3 + 31/2.- 4.6. The derived tower and the Coxeter invariant.- 4.7. Examples of derived towers.- 4.7.a. Finite group actions.- 4.7.b. The An Coxeter graphs.- 4.7.c. A general method.- 4.7.d. Some examples of derived towers for index 4 subfactors.- 4.7.e. The tunnel construction.- 4.7.f. The derived tower for R ? R?, when ? > 4.- Appendix I. Classification of Coxeter graphs with spectral radius just beyond the Kronecker range.- I.1. The results.- I.2. Computations of characteristic polynomials for ordinary graphs.- I.3. Proofs of theorems I.1.2 and I.1.3.- Appendix II.a. Complex semisimple algebras and finite dimensional C*-algebras.- Appendix III. Hecke groups and other subgroups of PSL(2,?) generated by parabolic pairs.- References.