Algebras of minimal rank over perfect fields

Let R(A) denote the rank (also called the bilinear complexity) of a finite-dimensional associative algebra A. A fundamental lower bound for R(A) is the so-called Alder-Strassen (1981) bound: R(A) ⩾ 2 dim A-t, where t is the number of maximal two-sided ideals of A. The class of algebras for which the Alder-Strassen bound is sharp, the so-called "algebras of minimal rank", has received wide attention in algebraic complexity theory. We characterize all algebras of minimal rank over perfect fields. This solves an open problem in algebraic complexity theory over perfect fields [as discussed by V. Strassen (1990) and P. Bürgisser et al. (1997)]. As a by-product, we determine all algebras A of minimal rank with A/rad A ≅ k^t over arbitrary fields