## Doctoral Theses

### Hypergroup Graphs and Subfactors.

2-28-1993

2-28-1994

#### Institute Name (Publisher)

Indian Statistical Institute

Doctoral Thesis

#### Degree Name

Doctor of Philosophy

Mathematics

#### Department

Theoretical Statistics and Mathematics Unit (TSMU-Bangalore)

Sunder, V. S.

#### Abstract (Summary of the Work)

The main theme of this t hesis is hypergroups. In this thesis the the- ory of hypergroups is applied to study the relation between certain graphs and subfactors of II, factors in the context of principal graphs associated with the inclusions of II, factors. More general classes of hypergroups are iutroduced, new examples of hypergroups associated to certain graphs are coustructed and classification of small order hypergroups is discussed.The text of the thesis is arranged in four chapters. The first chapter is on preliminaries of the theory of hypergroups, the second on the appli- cation of the theory of hyjrrgroups in the relation bet ween certain graphs and subfactors of II act.cs. the third on a more general class of hyper- groups and the fourth chapter is on some new examples of hypergroups and classification of smai order hypergroups.The first chapter on the preliminaries of the theory of hypergroups col- ts together the basic known facts about hypergroups which also serves the purpose of fixing notation and terminology for the following chapters. In this chapter the bimodule interpretation as against the relative commmitant interpretation of a principal graph associated with the inclusion of a of II, factors is worked out in detail.The second chapter is on the notion of an action of a hypergroup on a *et. After deriving some consequences of the definition of action, the notion is used here to show that certain bipartite graplÄ±s cannot arise as principal graphs for inclusions of II factors.The third chapter is on the notion of an Mrgraded hypergroup. This otion extends the notion of a hypergroup and captures the algebraic struc- ture pussesseed by the collection of irreducible bifinite bimodules over a pair of II, factors with respect to taking tensor products and contragredi- ents. The notion of a dimeusion function of a hypergroup is extended to M-graded hypergroups and it is proved that every irreducible finite M2- graded hypergroup possesSes a unique dincusion function. The results in this chapter also rule out some graphs from arising as principal graphs for inclusions of II1 factors.The fourth chapter is on some new examples of hypergroups and classi- fication of hypergroups of small order. Sequences of bypergroups associated to the graphs 32, for all positive integers n and the Coxeter graph E. for all positive integers except 7 and 10 are described here. More examples given by coutaected sum of certain graphs are also described here. This chapter coucludes with classification of hypergroups of small order which -hows that the smallest non-abelian hypergroup is the smallest non-abelian group.

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