Nil algebras with restricted growth

Abstract

It is shown that over an arbitrary countable field, there exists a finitely generated algebra that is nil, infinite dimensional, and has Gelfand-Kirillov dimension at most three.2010 Mathematics subject classification: 16N40, 16P90

On the Kurosh problem for algebras of polynomial growth over a general field

“…Proof of Proposition 3.1. The proof is similar to that of Lenagan and Smoktunowicz and the second-named author [LSY,Theorem 3.1]. We use induction on n and divide the proof into three cases.…”

“…In recent years there has been renewed interest in the construction of finitely generated algebraic algebras that are not finite-dimensional [BS,BSS,LS,LSY,Sm1,Sm2,Sm3]. The first such examples were constructed by Golod and Sharafervich [Go, GS] using a combinatorial criterion that guaranteed that algebras with certain presentations are infinitedimensional.…”

Matrix wreath products of algebras and embedding theorems

“…Suppose now that the algebra A is stable nil, the ground field F is countable, and GK dim A ≤ d. Let B be the Lenagan-Smoktunowicz-Young ( [16], [17]) nil algebra, GK dim B ≤ 3. Arguing as above, we see that the finitely generated algebra C, in which the algebra A is embedded, is nil.…”

“…Everywhere in this section, we assume that the ground field F is countable. Let B be an infinite dimensional graded finitely generated Lenagan-Smoktunowicz-Young nil algebra of Gelfand-Kirillov dimension ≤ 3 ( [16], [17]). Without loss of generality, we will assume that ℓ(B) = {b ∈ B|bB = (0)} = (0).…”

On graded characterizations of finite dimensionality for algebraic algebras

“…Since the early 2000s there has been increased interest -and several new significant results -in the study of Kurosh-type and related problems for associative algebras; see, e.g., [3], [16], and [18] for an introduction and overview. A thumbnail sketch of relevant developments could include: Smoktunowicz's 2002 construction of a simple nil ring over an arbitrary countable field [15]; Bell and Small's 2002 construction of a finitely generated primitive algebraic algebra over an arbitrary field [2]; Lenagan and Smoktunowicz's 2007 construction of an infinite dimensional nil algebra, over an arbitrary countable field, of finite Gelfand-Kirillov (GK-) dimension [9]; Lenagan, Smoktunowicz, and Young's 2012 construction of an infinite dimensional nil algebra, over an arbitrary countable field, of GK-dimension at most three [10]; and Bell, Small, and Smoktunowicz's construction of an infinite dimensional primitive algebraic algebra, over an arbitrary countable field, of GK-dimension at most six [3].…”