A lower bound for the bilinear complexity of some semisimple lie algebras

Abstract

Let U,V,W be finite dimensional vector spaces over a field k and let : U x V ~ W be a bilinear mapping. The (multiplicative) complexity L(@) of # is defined as the least rE I~ such that there are linear forms u 1,...,u r,v 1,...,v r 6 (UxV)* and elements w 1,...,wr6 W satisfying r #(x,y) = Z up(x,y) vp(x,y) wp for all (x,y)E U xV. p=1 The r-tuple ((up,vp,wp)6(UxV)*x(UxV)*x W , I <p =<r) is then called an (optimal) quadratic algorithm for ~. L(#) is the number of non linear arithmetic operations that are necessary and sufficient to compute @(x,y) from x,y by a straightline program (cf. Strassen (1973) and de Groote (1984) for further details). We shall use a somewhat coarser but more feasible computational model : A bilinear algorithm for ~ is a finite r-tuple