Identities of bilinear mappings and graded polynomial identities of matrices

We describe the polynomial identities of the bilinear mappings V W→K and Vr Vr*→Mr(K), where V, W, Vr are finite dimensional vector spaces over a field K of characteristic 0, dimVr=r and Mr(K) is the r×r matrix algebra. We show that these identities follow from the commutativity of the values in K of the bilinear forms V W→K and Vr* Vr→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 G are such that (g−1h)2≠e. In particular, we describe the graded identities depending only on the variables xi,h−1gxi,g−1h, i=1,2,…