The centre of generic algebras of small PI algebras

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• 作者：Plamen Koshlukov ; ¹ ; ^{plamen@ime.unicamp.br} ; Thiago Castilho de Mello² ; ^{tcmello@ime.unicamp.br}
• 关键词：16R10 ; 16R20 ; 16R99
• 刊名：Journal of Algebra
• 出版年：2013
• 出版时间：1 February, 2013
• 年：2013
• 卷：375
• 期：Complete
• 页码：109-120
• 全文大小：209 K

文摘

Verbally prime algebras are important in PI theory. They are well known over a field K of characteristic zero: 0 and (the trivial ones), , , . Here is the free associative algebra with free generators T, E is the infinite dimensional Grassmann algebra over K, and are the matrices over K and over E, respectively. Moreover are certain subalgebras of , defined below. The generic algebras of these algebras have been studied extensively. Procesi gave a very tight description of the generic algebra of . The situation is rather unclear for the remaining nontrivial verbally prime algebras.

In this paper we study the centre of the generic algebra of 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.