Filtrations and Distortion in Infinite-Dimensional Algebras

TL;DR

This paper explores how tame filtrations in infinite-dimensional algebras relate to embeddings and distortions, analyzing whether such filtrations can be derived from larger algebra structures, with implications for algebraic complexity.

Contribution

It investigates the conditions under which tame filtrations of algebras can be induced from degree filtrations of larger algebras, especially in associative and Lie algebras.

Findings

Tame filtrations can sometimes be induced from larger algebra filtrations.

Distortion reflects the complexity of membership problems in subalgebras.

Connections between algebraic distortion and geometric analogues are established.

Abstract

A tame filtration of an algebra is defined by the growth of its terms, which has to be majorated by an exponential function. A particular case is the degree filtration used in the definition of the growth of finitely generated algebras. The notion of tame filtration is useful in the study of possible distortion of degrees of elements when one algebra is embedded as a subalgebra in another. A geometric analogue is the distortion of the (Riemannian) metric of a (Lie) subgroup when compared to the metric induced from the ambient (Lie) group. The distortion of a subalgebra in an algebra also reflects the degree of complexity of the membership problem for the elements of this algebra in this subalgebra. One of our goals here is to investigate, mostly in the case of associative or Lie algebras, if a tame filtration of an algebra can be induced from the degree filtration of a larger algebra.