Central Polynomials For Z2-graded Algebras And For Algebras With Involution

We describe the Z2-graded central polynomials for the matrix algebra of order two, M2 (K), and for the algebras M1, 1 (E) and E ⊗ E over an infinite field K, char K ≠ 2. 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 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 Z2-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. © 2006 Elsevier Ltd. All rights reserved.2083877886Anisimov, N., Zp-codimensions of Zp-identities of Grassmann algebra (2001) Comm. Algebra, 29 (9), pp. 4211-4230. , (Special issue dedicated to Alexei Ivanovich Kostrikin)Azevedo, S.S., Fidelis, M., Koshlukov, P., Tensor product theorems in positive characteristic (2004) J. Algebra, 276 (2), pp. 836-845Azevedo, S.S., Fidelis, M., Koshlukov, P., Graded identities and PI equivalence of algebras in positive characteristic (2005) Comm. Algebra, 33 (4), pp. 1011-1022Belov, A., No associative PI algebra coincides with its commutant (2003) Siberian Math. J., 44 (6), pp. 969-980Colombo, J., Koshlukov, P., Central polynomials in the matrix algebra of order two (2004) Linear Algebra Appl., 377, pp. 53-67Colombo, J., Koshlukov, P., Identities with involution for the matrix algebra of order two in characteristic p (2005) Israel J. Math., 146, pp. 337-355Di Vincenzo, O.M., On the graded identities of M1, 1 (E) (1992) Israel J. Math., 80 (3), pp. 323-335Formanek, E., Central polynomials for matrix rings (1972) J. Algebra, 23, pp. 129-132Formanek, E., Invariants and the ring of generic matrices (1984) J. Algebra, 89, pp. 178-223Giambruno, A., Koshlukov, P., On the identities of the Grassmann algebras in characteristic p > 0 (2001) Israel J. Math., 122, pp. 305-316Kemer, A., (1991) Translations Math. Monographs, 87. , Amer. Math. Soc., Providence, RIKoshlukov, P., Basis of the identities of the matrix algebra of order two over a field of characteristic p ≠ 2 (2001) J. Algebra, 241, pp. 410-434Koshlukov, P., Azevedo, S.S., Graded identities for T-prime algebras over fields of positive characteristic (2002) Israel J. Math., 128, pp. 157-176Krakowski, D., Regev, A., The polynomial identities of the Grassmann algebra (1973) Trans. Amer. Math. Soc., 181, pp. 429-438Levchenko, D., Finite basis property of identities with involution of a second-order matrix algebra (1982) Serdica, 8 (1), pp. 42-56. , (in Russian)Okhitin, S., Central polynomials of the algebra of second order matrices (1988) Moscow Univ. Math. Bull., 43 (4), pp. 49-51Popov, A., Identities of the tensor square of a Grassmann algebra (1982) Algebra Logic, 21, pp. 296-316Razmyslov, Yu.P., Finite basing of the identities of a matrix algebra of second order over a field of characteristic zero (1973) Algebra Logic, 10, pp. 47-63Razmyslov, Yu., (1994) Translations Math. Monographs, 138. , AMS, Providence, RIRowen, L.H., (1980) Polynomial Identities in Ring Theory, , Acad. Press, N