# Homogeneous integral table algebras of degree three: a trilogy

### About this Title

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

Publication: Memoirs of the American Mathematical Society

Publication Year
2000: Volume 144, Number 684

ISBNs: 978-0-8218-2021-6 (print); 978-1-4704-0275-4 (online)

DOI: http://dx.doi.org/10.1090/memo/0684

MathSciNet review: 1672250

MSC: Primary 20C99

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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.

Readership

Graduate students and research mathematicians interested in
commutative rings and algebras.

### Table of Contents

**Chapters**

- I. Homogeneous integral table algebras of degree three with a faithful real
element
- II. On antisymmetric homogeneous integral table algebras of degree three
- III. Homogeneous integral table algebras of degree three with no nontrivial
linear elements