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The multiplication algebra of a train algebra of rank 3. (English) Zbl 0965.17020

From the text: The authors introduce the multiplication algebra of a train algebra of rank 3 and study its basic properties: Idempotents, Peirce decomposition, some numerical invariants, maximal ideals, etc.
They prove the following Theorem 1: Let \(A= Ke\oplus U\oplus V\) and \(T_0\) the idempotent of (the multiplication algebra) \(M(A)\) defined by \(T_0(e)= e\) and \(T_0|_N= 0\). Then:
(a) \(M(A)= KT_0 \oplus \widetilde{U} \oplus \widetilde{V}\), where \(\widetilde{U}= \{\sigma\in (N:A): \sigma T_0= \sigma\}\), \(\widetilde{V}= \{\sigma\in (N:A): \sigma T_0= 0\}\) and \((N:A)= \widetilde{U} \oplus \widetilde{V}\).
(b) The subspaces \(\widetilde{U}\) and \(\widetilde{V}\) can be characterized by the following conditions on \(\sigma\in (N:A)\): \(\sigma\in \widetilde{U}\) if and only if \(\sigma(N)= 0\) and \(\sigma\in \widetilde{V}\) if and only if \(\sigma(e)= 0\).
(c) The following inclusions hold: \(\widetilde{U}^2= 0\), \(\widetilde{U} \widetilde{V}= 0\), \(\widetilde{V} \widetilde{U} \subseteq \widetilde{U}\), \(\widetilde{V}^2 \subseteq \widetilde{V}\).
(d) \(\widetilde{U}\) is an ideal of zero square and \(\widetilde{V}\) is a subalgebra of \(M(A)\).
Theorem 2: Let \(A= Ke \oplus U\oplus V\) where \(U\) and \(V\) are nonzero and \(N= U\oplus V\) is nilpotent. Then the unique maximal ideals of \(M(A)\) are the ideals \(M_0\), \(M_1\) and \(M_2\) defined in the paper.

MSC:

17D92 Genetic algebras
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