Algebras of minimal rank over perfect fields

Abstract

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) /spl ges/ 2 dim A-t, where t is the number of maximal two-sided ideals of A. The class of 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 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. Bu/spl uml/rgisser et al. (1997)]. As a by-product, we determine all A of minimal rank with A/rad A /spl cong/ k/sup t/ over arbitrary fields.