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arXiv · 1208.1317

The centre of generic algebras of small PI algebras

Thiago Castilho de Mello·Plamen Koshlukov
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Abstract

Verbally prime algebras are important in PI theory. They are well known 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 with free generators $T$, $E$ is 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. Moreover $M_{ab}(E)$ are certain subalgebras of $M_{a+b}(E)$, defined below. The generic algebras of these algebras have been studied extensively. Procesi gave a very tight description of the generic algebra of $M_n(K)$. The situation is rather unclear for the remaining nontrivial verbally prime algebras. In this paper we study the centre of the generic algebra of $M_{11}(E)$ in two generators. We prove that this centre is a direct sum of the field and a nilpotent ideal (of the generic algebra). We describe the centre of this algebra. As a corollary we obtain that this centre contains nonscalar elements thus we answer a question posed by Berele.

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BibTeXRIS

Thiago Castilho de Mello, Plamen Koshlukov. 2012-11-16. The centre of generic algebras of small PI algebras. https://doi.org/10.1016/j.jalgebra.2012.11.018

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