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Standard integral table algebras with a faithful nonreal element of degree 5
Publikationstyp
Book Part
Date Issued
2002
Sprache
English
Author(s)
Institut
TORE-URI
First published in
Number in series
1773 LNM
Start Page
43
End Page
81
Citation
Standard integral table algebras generated by a non-real element of small degree / Zvi Arad; Mikhail Muzychuk (eds.). - Berlin ; Heidelberg [u.a.] : Springer, 2002. - VII, 126 S. ; 24 cm ISBN 3-540-42851-8 kart. : DM 45.90 (Lecture notes in mathematics ; 1773). - Seite 43-81
Publisher DOI
Publisher
Springer
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 λxyz|z|⟨xy,z⟩=⟨x,zy¯⟩=λzy¯x|x|,x,y,z∈B,
one can list all possible representations of bb¯ and b 2 as linear combinations of basis elements (cf. Tables II and III of Subsection 3.3). Assuming that b commutes with b¯ yields the identity ⟨bb¯,bb¯⟩=⟨b2,b2⟩ 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.
one can list all possible representations of bb¯ and b 2 as linear combinations of basis elements (cf. Tables II and III of Subsection 3.3). Assuming that b commutes with b¯ yields the identity ⟨bb¯,bb¯⟩=⟨b2,b2⟩ 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.
DDC Class
510: Mathematik