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arXiv · 1208.2185

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

Thiago Castilho de Mello·Plamen Koshlukov
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Abstract

Verbally prime algebras are important in PI theory. They were described by Kemer over a field $K$ of characteristic zero: 0 and $K $ (the trivial ones), $M_n(K)$, $M_n(E)$, $M_{ab}(E)$. Here $K $ is the free associative algebra of infinite rank, with free generators $T$, $E$ denotes the infinite dimensional Grassmann algebra over $K$, $M_n(K)$ and $M_n(E)$ are the $n\times n$ matrices over $K$ and over $E$, respectively. The algebras $M_{ab}(E)$ are subalgebras of $M_{a+b}(E)$, see their definition below. The generic (also called relatively 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 but their 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 centre is 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 obtained assuming the base field infinite and of characteristic different from 2. In this paper we study the polynomial identities satisfied by this generic algebra. We exhibit a basis of its polynomial identities. It turns out that this algebra is PI equivalent to a 5-dimensional algebra of certain upper triangular matrices. The identities of the latter algebra have been studied; these were described by Gordienko. As an application of our results we describe the subvarieties of the variety of unitary algebras generated by the generic algebra in two generators of $M_{11}(E)$. Also we describe the polynomial identities in two variables of the algebra $M_{11}(E)$.

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Thiago Castilho de Mello, Plamen Koshlukov. 2012-08-10. On the polynomial identities of the algebra $M_{11}(E)$. https://doi.org/10.1016/j.laa.2013.01.032

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