Primitive Algebras with Arbitrary Gelfand-Kirillov Dimension

We construct, for every real β ≥ 2, a primitive affine algebra with Gelfand-Kirillov dimension β. Unlike earlier constructions, there are no assumptions on the base field. In particular, this is the first construction over or L. Given a recursive sequence {vn} of elements in a free monoid, we investigate the quotient of the free associative algebra by the ideal generated by all nonsubwords in {vn}. We bound the dimension of the resulting algebra in terms of the growth of {vn}. In particular, if νn is less than doubly exponential, then the dimension is 2. This also answers affirmatively a conjecture of Salwa (1997, Comm. Algebra 25, 3965–3972).