Lower bounds for the bilinear complexity of associative algebras

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Let R(A) denote the bilinear complexity (also called rank) of a finite dimensional associative algebra A.¶We prove that $ R(A) \ge {5 \over 2} {\rm dim}\,A - 3({n_1}+\cdots+{n_t}) $ if the decomposition of $ A/{\rm rad}\,A \cong {A_1} \times \cdots \times {A_t} $ into simple algebras $ {A_\tau} \cong D^{n_\tau\times n_\tau}_{\tau} $ contains only noncommutative factors, that is, the division algebra $ D_\tau $ is noncommutative or $ n_\tau \ge 2 $ . In particular, $ n \times n $ -matrix multiplication requires at least $ {5\over2}n^2 - 3n $ essential bilinear multiplications. We also derive lower bounds of the form $ {5 \over 2}n^2 - 3n $ essential bilinear multiplications. We also derive lower bounds of the form $ {5 \over 2} - o(1)) \cdot {\rm dim}\,A $ for the algebra of upper triangular $ n \times n $ -matrices and the algebra $ k[X,Y]/(X^{n+1}, X^{n}Y, X^{n-1}Y^2,\cdots,Y^{n+1}) $ of truncated bivariate polynomials in the indeterminates X,Y over some field k.¶A class of algebras that has received wide attention in this context con-sists of those algebras A for which the Alder—Strassen Bound is sharp, i.e., R(A) = 2dim A—t is the number of maximal twosided ideals in A. These algebras are called algebras of minimal rank. We determine all semisimple algebras of minimal rank over arbitrary fields and all algebras of minimal rank over algebraically closed fields.