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Homogeneous Integral Table Algebras of Degree Three: A Trilogy
Harvey I. Blau and Bangteng Xu, Northern Illinois University, DeKalb, IL, and Z. Arad, E. Fisman, V. Miloslavsky, and M. Muzychuk, Bar-Ilan University, Ramat-Gan, Israel
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Memoirs of the American Mathematical Society
2000; 89 pp; softcover
Volume: 144
ISBN-10: 0-8218-2021-4
ISBN-13: 978-0-8218-2021-6
List Price: US$51 Individual Members: US$30.60
Institutional Members: US\$40.80
Order Code: MEMO/144/684

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.

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

Part I. Homogeneous Integral Table Algebras of Degree Three with a Faithful Real Element, H. I. Blau and B. Xu
• Introduction
• Known facts and some consequences
• Homogeneous ITA's of arbitrary degree
• Some results on bases with a standard quotient
• Extensions of $$\mathbf T_n(3), n>1$$
• Extensions of $$\mathbf V_3$$
• Extensions of $$\mathbf V_2$$ and $$\mathbf V_4$$
• Extensions of $$\mathbf T_0(3)$$
• Proof of the main theorem
• References
Part II. On Antisymmetric Homogeneous Integral Table Algebras of Degree Three, Z. Zrad, E. Fisman, V. Miloslavsky, and M. Muzychuk
• Introduction
• General facts
• The universal covering
• Perfect triples
• References
Part III. Homogeneous Integral Table Algebras of Degree Three With No Nontrivial Linear Elements, H. I. Blau
• Introduction
• Tame and wild elements
• The array
• The cover
• Proofs of Theorem A and Corollary C
• Association schemes
• References