Central polynomials for [formula omitted]-graded algebras and for algebras with involution

Abstract

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