On bilinear algorithms for multiplication in quaternion algebras

We show that the bilinear complexity of multiplication in a non-split quaternion algebra over a field of characteristic distinct from 2 is 8. This question is motivated by the problem of characterising algebras of almost minimal rank studied in [1] This paper is a translation of a report submitted by the author to the XI international seminar ”Discrete mathematics and its applications”. Definition 1. A sequence (f1, g1, z1; . . . ; fr, gr, zr) with fk ∈ U , gk ∈ V , zk ∈ W is called a bilinear algorithm of length r for a bilinear mapping φ : U × V → W if