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

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 b ² 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.