## Doctoral Theses

### Connections on Small Vertex Model.

2-28-1988

2-28-1989

#### Institute Name (Publisher)

Indian Statistical Institute

Doctoral Thesis

#### Degree Name

Doctor of Philosophy

Computer Science

#### Department

Theoretical Statistics and Mathematics Unit (TSMU-Kolkata)

Sunder, V. S.

#### Abstract (Summary of the Work)

This thesis is devoted to the classification of a special class of commuting squares called vertex models.The first chapter is indroductory in nature and is included for the sake of completeness and convenience of reference. It starts with the description of the basic construction and the invariants called the principal and dual graph for an inclusion of II1, factors. After defining a commuting square we describe the special class of commuting squares called vertex model given by an Mn, O Mk, biunitary matrix . Finally we state some results, without proof, about vertex nodels from (KSC).The second chapter in the core of the thesis. It is devoted to the classification of vertex models given by an Mn, O M2, biunitary matrix. If B(k, n) denotes the collection of Mn. M2, biunitary matrices, then B(2,3) is classified up to the nat- ural equivalence relation on it. Further, a simple model form for a representative from each equivalence class in B(2, n) and also neceasary conditions for two such model connections' to be themselves equivalent are obtained. Then we go on to show that B(2, n) contains a (3n - 6) parameter family of pairwise inequivalent connections and show that the number (3n - 6) is sharp. Finally, it is deduced that every graph that can arise as the principal graph of a finite depth subfactor of index 4 actually arises for one arising from a vertex model corresponding to B(2, n) for some n.In the appendix, we give elementary and direct proofs of two known results in the literature (due to Kosaki-Yamagami and Bisch, respectively) using the techniques of bimodules and elementary matrix manipulations. The first result is the computation of principal and dual graphs for the inclusion of Il, factors N - P gt;4H CP gt;4G = M, where G is a discrete group acting as outer automorphisms of a I, factor P, and H is a subgroup of G such that (G : H) < o. It is this proof that has basically been reproduced in (JS). The second result (due to Bisch) is that if NC MCPis an inclusion of I1, factors such that NC P has finite depth, then NCM and MCP have finite depth.

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