Primitive algebraic algebras of polynomially bounded growth. (Q2895454)

In recent years, considerable progress has been made in constructing finitely generated algebras of small polynomially bounded growth that are algebraic and infinite-dimensional over their base fields. In a paper by \textit{T.~H.~Lenagan, A.~Smoktunowicz}, and \textit{A.~A. Young}, an example of such an algebra over a countable field is provided that has Gelfand-Kirillov dimension (\(\text{GK}\dim\)) at most three [Proc. Edinb. Math. Soc., II. Ser. 55, No. 2, 461-475 (2012; Zbl 1259.16022)].NEWLINENEWLINE However, in most of these constructions, the Jacobson radical is large and nil, so they contribute very little to the question whether or not there exists an algebraic division ring that is finitely generated and infinite-dimensional over its centre. As a partial step towards resolving this problem, the authors of the article under review look at primitive algebras, and by using a technique called affinization, they establish, for a countable field \(k\), the existence of a finitely generated infinite-dimensional algebraic primitive \(k\)-algebra \(A\) with \(\text{GK}\dim(A)\leq 6\). Another affinization construction provides a two-generator infinite-dimensional primitive algebraic \(k\)-algebra. The paper ends with eight questions that are related to the above results.NEWLINENEWLINEFor the entire collection see [Zbl 1232.16001].