Central polynomials for Z2Z2-graded algebras and for algebras with involution

We describe the Z2Z2-graded central polynomials for the matrix algebra of order two, M2(K)M2(K), and for the algebras M1,1(E)M1,1(E) and E⊗EE⊗E over an infinite field KK, charK≠2. Here EE is the infinite-dimensional Grassmann algebra, and M1,1(E)M1,1(E) stands for the algebra of the 2×22×2 matrices whose entries on the diagonal belong to E0E0, the centre of EE, and the off-diagonal entries lie in E1E1, the anticommutative part of EE. It turns out that in characteristic 0 the graded central polynomials for M1,1(E)M1,1(E) and E⊗EE⊗E are the same (it is well known that these two algebras satisfy the same polynomial identities when charK=0). On the contrary, this is not the case in characteristic p>2p>2. We describe systems of generators for the Z2Z2-graded central polynomials for all these algebras.Finally we give a generating set of the central polynomials with involution for M2(K)M2(K). We consider the transpose and the symplectic involutions.