An infinite dimensional affine nil algebra with finite Gelfand-Kirillov dimension

Abstract

The famous 1960’s construction of Golod and Shafarevich yields infinite dimensional nil, but not nilpotent, algebras. However, these algebras have exponential growth. Here, we construct an infinite dimensional nil, but not locally nilpotent, algebra which has polynomially bounded growth.

Nil algebras with restricted growth

“…The proof of the first claim is very similar to the proof of [5,Theorem 9] and so is omitted. Notice that,…”

“…Suprisingly, this is not the case: in [5] Lenagan and Smoktunowicz constructed, over any countable field, an infinite dimensional finitely generated nil algebra with Gelfand-Kirillov dimension at most 20. This result raises the following question: what is the minimal rate of growth for a finitely generated infinite dimensional nil algebra?…”

Combinatorial Algebra: Syntax and Semantics

“…Now take t = [199]. In fact they managed to construct a finitely generated nil-algebra over any countable field given by homogeneous polynomial relations such that the algebra has polynomial growth, i.e., its Gelfand-Kirillov dimension is finite.…”

Some results in noncommutative ring theory

“…A nil element is obviously algebraic, and, in the converse direction, it is possible to construct an associated graded algebra connected with an algebraic algebra in such a way that the positive part is graded nil, i.e., all homogeneous elements are nil. On the other hand, the Kurosh Problem has a negative solution for rings with finite Gelfand-Kirillov dimension ( [30]), for simple rings ( [42]), for primitive rings ( [2]), for finitely generated primitive rings ( [8]), and for finitely generated algebraic primitive rings ( [9]). However, a natural question arising from the general Kurosh Problem remains open: Question 4 (Small's question).…”