On the algebras of almost minimal rank

V. V. Lysikov


We consider the bilinear complexity of multiplication in local and semisimple algebras over an infinite field of characteristic differing from 2. We obtain a criterion for the rank of a local algebra to be almost minimal. We evaluate the bilinear complexity of the algebras of generalised quaternions over such a field; we prove that any simple algebra of almost minimal rank is an algebra of generalised quaternions. This result is used for the classification of semisimple algebras of almost minimal rank.

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