Identities of bilinear mappings and graded polynomial identities of matrices

We describe the polynomial identities of the bilinear mappings V circle times W --> K and V-r circle times Vr* --> M-r(K), where V, W, V-r are finite dimensional vector spaces over a field K of characteristic 0, dim V-r = r and M-r(K) is the r x r matrix algebra. We show that these identities follow from the commutativity of the values in K of the bilinear forms V circle times W --> K and V-r* circle times V-r --> K and Capelli identities. We apply these results to the G-graded polynomial identities of the algebra Mr(K) with the elementary G-grading associated with the r-tuple (g,.... g, h), where G is an arbitrary group and g, h is an element of G are such that (g(-1) h)(2) not equal e. In particular, we describe the graded identities depending only on the variables x(i,h-1g)x(i,g-1h), i = 1, 2.... (C) 2003 Elsevier Science Inc. All rights reserved.