Algebras of Minimal Rank over Arbitrary Fields

Abstract

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) ≥ 2 dim A-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.As the main contribution of this work, we characterize all algebras of minimal rank over arbitrary fields. This finally solves an open problem in algebraic complexity theory, see for instance [], Sect. 12, Problem 4] or [], Problem 17.5].