Standard Integral Table Algebras with a Faithful Nonreal Element of Degree 5

  • Z. Arad
  • F. Bünger
  • E. Fisman
  • M. Muzychuk
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 1773)

Abstract

This chapter deals with the classification of standard integral GT-algebras (A,B) with L(B) = 1 {1} and |b| ≥ 4 for all b ∈ B# which contain a nonreal faithful basis element b of degree 5. Starting from this point using the basic identity
$$ \lambda _{xyz} |z|\left\langle {xy,z} \right\rangle = \left\langle {x,z\bar y} \right\rangle = \lambda _{z\bar yx} |x|,x,y,z \in B, $$
one can list all possible representations of \( b\bar b \) and b2 as linear combinations of basis elements (cf. Tables II and III of Subsection 3.3). Assuming that b commutes with \( \bar b \) yields the identity \( \left\langle {b\bar b,b\bar b} \right\rangle = \left\langle {b^2 ,b^2 } \right\rangle \) which reduces the number of these representations (cf. Table III of Subsection 3.3). Then, using various kind of techniques (for example repeated application of the associa- tivity law), each of the reamining cases will be treated separately. In order to state the main result, we introduce the following base of a specific table algebra.

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Z. Arad
    • 1
    • 2
  • F. Bünger
    • 1
  • E. Fisman
    • 1
  • M. Muzychuk
    • 1
    • 2
  1. 1.Department of Mathematics and Computer ScienceBar-Ilan UniversityRamat-GanIsrael
  2. 2.Department of Mathematics and Computer ScienceNetanya Academic CollegeNetanyaIsrael

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