A Complete Characterization of the Algebras of Minimal Bilinear Complexity

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) \ge 2 \dim A - t, where t is the number of maximal twosided ideals of A. An algebra is called an algebra of minimal rank if the Alder--Strassen bound is tight, i.e., $R(A) = 2 \dim A - t. 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, [V. Strassen, Handbook of Theoretical Computer Science, J. van Leeuwen, ed., Elsevier Science, New York, 1990, Vol. A, pp. 634--672, section 12, Problem 4] or [P. Burgisser, M. Clausen, and M. A. Shokrollahi, Algebraic Complexity Theory, Springer, New York, 1997, Problem 17.5].