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Verbally prime algebras are important in PI theory. They were described byKemer over a field $K$ of characteristic zero: 0 and $K<T>$ (the trivial ones),$M_n(K)$, $M_n(E)$, $M_{ab}(E)$. Here $K<T>$ is the free associative algebra ofinfinite rank, with free generators $T$, $E$ denotes the infinite dimensionalGrassmann algebra over $K$, $M_n(K)$ and $M_n(E)$ are the $n\times n$ matricesover $K$ and over $E$, respectively. The algebras $M_{ab}(E)$ are subalgebrasof $M_{a+b}(E)$, see their definition below. The generic (also calledrelatively free) algebras of these algebras have been studied extensively.Procesi described the generic algebra of $M_n(K)$ and lots of its properties.Models for the generic algebras of $M_n(E)$ and $M_{ab}(E)$ are also known buttheir structure remains quite unclear.  In this paper we study the generic algebra of $M_{11}(E)$ in two generators,over a field of characteristic 0. In an earlier paper we proved that its centreis a direct sum of the field and a nilpotent ideal (of the generic algebra),and we gave a detailed description of this centre. Those results were obtainedassuming the base field infinite and of characteristic different from 2. Inthis paper we study the polynomial identities satisfied by this genericalgebra. We exhibit a basis of its polynomial identities. It turns out thatthis algebra is PI equivalent to a 5-dimensional algebra of certain uppertriangular matrices. The identities of the latter algebra have been studied;these were described by Gordienko. As an application of our results we describethe subvarieties of the variety of unitary algebras generated by the genericalgebra in two generators of $M_{11}(E)$. Also we describe the polynomialidentities in two variables of the algebra $M_{11}(E)$.

On the polynomial identities of the algebra $M_{11}(E)$