Semisimple algebras of almost minimal rank over the reals

A famous lower bound for the bilinear complexity of the multiplication in associative algebras is the Alder–Strassen bound. Algebras for which this bound is tight are called algebras of minimal rank. After 25 years of research, these algebras are now well understood. Here we start the investigation of the algebras for which the Alder–Strassen bound is off by one. As a first result, we completely characterize the semisimple algebras over R whose bilinear complexity is by one larger than the Alder–Strassen bound. Furthermore, we characterize all algebras A (with radical) of minimal rank plus one over R for which A/radA has minimal rank plus one. The other possibility is that A/radA has minimal rank. For this case, we only present a partial result.