# Homogeneous Integral Table Algebras of Degree Three: A Trilogy

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*Harvey I. Blau; Bangteng Xu; Z. Arad; E. Fisman; V. Miloslavsky; M. Muzychuk*

Homogeneous integral table algebras of degree three with a
faithful real element. The algebras of the title are classified to exact
isomorphism; that is,
the sets of structure constants which arise from the given basis are
completely determined. Other results describe all possible extensions
(pre-images), with a faithful element which is not necessarily real, of
certain simple homogeneous integral table algebras of degree three.

On antisymmetric homogeneous integral table algebras of
degree three. This paper determines the homogeneous integral table algebras of
degree
three in which the given basis has a faithful element and has no
nontrivial elements that are either
real (symmetric) or linear, and where an additional hypothesis is
satisfied. It is shown that all such bases must occur as the set of
orbit sums in the complex group algebra of a finite abelian group under
the action of a fixed-point-free automorphism of order three.

Homogeneous integral table algebras of degree three with
no nontrivial linear elements. The algebras of the title which also have a
faithful element are
determined to exact isomorphism. All of the simple homogeneous integral
table algebras of degree three are displayed, and the commutative
association schemes in which all the nondiagonal relations have valency
three and where some relation defines a connected graph on the
underlying set are classified up to algebraic isomorphism.

#### Table of Contents

# Table of Contents

## Homogeneous Integral Table Algebras of Degree Three: A Trilogy

- Contents vii8 free
- Preface ix10 free
- Part I. Homogeneous Integral Table Algebras of Degree Three with a Faithful Real Element 114 free
- 1. Introduction 114
- 2. Known Facts and Some Consequences 1124
- 3. Homogeneous ITA's of Arbitrary Degree 1528
- 4. Some Results on Bases with a Standard Quotient 2033
- 5. Extensions of T[Sub(n)](3), n > 1 2942
- 6. Extensions of V[Sub(3)] 3245
- 7. Extensions of V[Sub(2)] and V[Sub(4)] 3851
- 8. Extensions of T[Sub(0)](3) 4558
- 9. Proof of the Main Theorem 5265
- References 5366

- Part II. On Antisymmetric Homogeneous Integral Table Algebras of Degree Three 5467
- Part III. Homogeneous Integral Table Algebras of Degree Three With No Nontrivial Linear Elements 7487