A new measure of growth for countable-dimensional algebras

Abstract

Abstract.A new dimension function on countable-dimensional algebras (over a field) is described. Its dimension values for finitely generated algebras exactly fill the unit interval [0,1 ]. Since the free algebra on two generators turns out to have dimension 0 (although conceivably some Noetherian algebras might have positive dimension!), this dimension function promises to distinguish among algebras of infinite GK-dimension.

Gromov translation algebras over discrete trees are exchange rings

“…The growth algebras G(r), introduced by Hannah and O'Meara [10,11,13] in the early 1990's, were originally defined over a field, but the same definitions still apply when the matrix entries are from any ring R. The ring of all ω × ω matrices over R which are simultaneously row-finite and column-finite is denoted by B(R). For a matrix x ∈ B(R), a growth curve for x is any function g :…”

“…This appears to be new even in the case where R is a field. In that setting, the papers [10,11,13] referred to the corresponding algebra of constant bandwidth matrices as the growth algebra G(0), because it was the first of a whole spectrum of growth algebras G(r), for r in the unit interval [0,1], that were used to define the bandwidth dimension of an arbitrary countable-dimensional algebra. For 0 < r ≤ 1, the G(r) are still translation algebras relative to a suitable pseudo-metric, and an easy extension (Theorem 3.1) of our main result shows that all the G(r) are exchange algebras.…”

“…We would like to thank John Roe for bringing translation algebras to the attention of the second author following the publication of [10,11,13]. …”

A new measure of growth for groups and algebras

“…In [HO1,HO2,O], Hannah and O'Meara introduced and studied a new notion of growth for algebras. This notion does not involve the growth of an algebra in terms of generators.…”

Row and column finite matrices

“…Recently it was shown that any countably generated algebra over a field K can be embedded in RCFM(K) [3,Proposition 2.1]. This result, in turn, was used to define a new dimension function on such algebras [5]. The ring RCFM(R) has arisen in the context of Morita equivalence [4], exchange rings [1,7], and Baer rings [2].…”