Central polynomials for image-graded algebras and for algebras with involution

We describe the image-graded central polynomials for the matrix algebra of order two, M2(K), and for the algebras M1,1(E) and Ecircle times operatorE over an infinite field K, image. Here E is the infinite-dimensional Grassmann algebra, and M1,1(E) stands for the algebra of the 2×2 matrices whose entries on the diagonal belong to E0, the centre of E, and the off-diagonal entries lie in E1, the anticommutative part of E. It turns out that in characteristic 0 the graded central polynomials for M1,1(E) and Ecircle times operatorE are the same (it is well known that these two algebras satisfy the same polynomial identities when image). On the contrary, this is not the case in characteristic p>2. We describe systems of generators for the image-graded central polynomials for all these algebras. Finally we give a generating set of the central polynomials with involution for M2(K). We consider the transpose and the symplectic involutions.