Distinguish subexponentially amenable and amenable Lie algebras

Determine whether the class of subexponentially amenable Lie algebras SL is strictly smaller than the class of amenable Lie algebras AL.

Background

The paper defines EL as the class generated by finite-dimensional and Abelian Lie algebras under subalgebras, quotients, extensions, and directed unions. It defines SL similarly, starting from Lie algebras of subexponential growth, and lets AL denote the class of all amenable Lie algebras. The inclusions EL SL AL are established.

References

We do not know, at present, whether \textsf{SL} and \textsf{AL} are distinct, as is the case for groups, but we can at least show (in a surprisingly difficult manner; Be'eri Greenfeld supplied us with a simpler construction based on):

— Amenability of Lie, Group and Hopf algebras  (2609.10373 - Bartholdi, 9 Sep 2026) in Introduction, paragraph following the definition of the classes EL, SL, and AL