Lower bounds for the bilinear complexity of associative algebras

Let R(A) denote the bilinear complexity (also called rank) of a finite dime nsional associative algebra A. We prove that R(A) Z greater than or equal to 5/2 dim A - 3(n(1) +...+ n(t) ) if the decomposition of A/rad A congruent to A(x) x...x A(t) into simple algebras A(tau) congruent to D-tau(n tau) congruent to D-tau(n tau xn tau) contains only noncommutative factors, that is, the division algebra D-tau i s noncommutative or n(tau) greater than or equal to 2. In particular, n x n -matrix multiplication requires at least 5/2n(2) - 3n essential bilinear mu ltiplications. We also derive lower bounds of the form (5/2 - o(1)).dim A f or the algebra of upper triangular n x n-matrices and the algebra k [X, Y]/ (Xn+1, (XY)-Y-n, (Xn-1Y2),..., Yn+1) of truncated bivariate polynomials in the indeterminates X, Y over some held k. A class of algebras that has received wide attention in this context consis ts of those algebras A for which the Alder-Strassen Bound is sharp, i.e., R (A) = 2 dim A - t, where t is the number of maximal twosided ideals in A. T hese algebras are called algebras of minimal rank. We determine all semisim ple algebras of minimal rank over arbitrary fields and all algebras of mini mal rank over algebraically closed fields.