Algebras of minimal rank: overview and recent developments

Let R(A) denote the rank (also called bilinear complexity) of a finite dimensional associative algebra A. A fundamental lower bound for R(A) is the so-called Alder-Strassen bound R(A) ≥ 2dimA − t, where t is the number of maximal twosided ideals of A. The class of algebras for which the Alder- Strassen bound is sharp, the so-called algebras of minimal rank, has received a wide attention in algebraic complexity theory. A lot of effort has been spent on describing algebras of minimal rank in terms of their algebraic structure. We review the known characterisation results, study some examples of algebras of minimal rank, and have a quick look at some central techniques. We put special emphasis on recent developments.