Free limits of free algebras

TL;DR

This paper investigates the limits of sequences of free algebraic structures, showing that under certain conditions, these limits preserve freeness and are explicitly describable in terms of generators.

Contribution

It establishes conditions under which limits of free algebras remain free, providing explicit descriptions for associative and Lie algebra cases.

Findings

Limits of free associative algebras are free graded algebras.

Limits of free associative algebras in ungraded case are free formal power series algebras.

Limits of free Lie algebras are free graded Lie algebras.

Abstract

Consider a diagram $\cdots \to F_3 \to F_2\to F_1$ of algebraic systems, where $F_n$ denotes the free object on $n$ generators and the connecting maps send the extra generator to some distinguished trivial element. We prove that (a) if the $F_i$ are free associative algebras over a fixed field then the limit in the category of graded algebras is again free on a set of homogeneous generators; (b) on the other hand, the limit in the category of associative (ungraded) algebras is a free formal power series algebra on a set of homogeneous elements, and (c) if the $F_i$ are free Lie algebras then the limit in the category of graded Lie algebras is again free.