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On train algebras of rank 3. (English) Zbl 0728.17019
First, the author recalls some concepts introduced in his preceding paper [cf. Proc. Edinb. Math. Soc., II. Ser. 33, No.1, 61-70 (1990; Zbl 0671.17022)]: train algebras of rank 3, the type of such algebras, which is an ordered pair of integers. In section 1 all train algebras of type (n,1) are classified such that dim J$$=\dim B_ 1B_ 2\oplus B_ 2\geq 2$$. In section 2 train algebras of type (3,n-2) such that dim J$$=n-1$$ are classified. In section 3 all train algebras $$B'$$ are considered whose kernel B is anisotropic, that is to say that dim $$B^ 2=1$$ and the bilinear form defined by the multiplication has a one-dimensional radical and its regular part is anisotropic.

##### MSC:
 17D92 Genetic algebras
##### Keywords:
genetic algebra; anisotropic kernel; train algebras
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##### References:
 [1] R. Costa, Train algebra of rank 3 and dimension ⩽ 5, Proc. Edinburgh Math. Soc., (to appear). · Zbl 0671.17022 [2] Scharlau, W., Quadratic and Hermitian forms, (1984), Springer-Verlag New York [3] Worz-Busekros, A., Algebras in genetics, Lecture notes in biomath., 36, (1980) · Zbl 0431.92017
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