The centre of generic algebras of small PI algebras. (Q372662)

The Grassmann algebra \(E=E_0\oplus E_1\) of an infinite dimensional vector space is graded by the \(2\)-element group. Denote by \(E_{1,1}\) the algebra of \(2\times 2\) matrices with diagonal entries from \(E_0\) and off-diagonal entries from \(E_1\). This algebra plays significant role in the theory of polynomial identities, since when the base field has characteristic zero, the T-ideal of identities of this algebra is verbally prime, and is the same as the T-ideal of identities of the tensor square of \(E\).NEWLINENEWLINE In the present paper the center of the rank two relatively free algebra of the variety generated by \(E_{1,1}\) is studied (over an infinite base field of characteristic different from \(2\)). A corollary of the calculations in the paper is that the center is the sum of the base field and a nilpotent ideal (of the relatively free algebra). The approach of the paper uses a realization of this relatively free algebra as an algebra of generic matrices with entries from the free supercommutative superalgebra.