Carpet Lie Rings Not Invariant with Respect to Carpet Subgroups

This article is a continuation of the author’s papers [Proc. Steklov Inst. Math., 2013; J. Sib. Fed. Univ. Math. Phys., 2023; Sib. Math. J., 2026]. In the first of them, necessary and sufficient conditions for a carpet Lie ring to be invariant with respect to a carpet subgroup were found, whereas in the second and third papers it was established that these conditions are sufficient for the original carpet to be closed. Here we construct examples of irreducible closed carpets over a field such that the carpet Lie rings are not invariant with respect to the corresponding carpet subgroups; at the same time, depending on the type of the root system associated with the carpet, some restrictions are imposed on the characteristic of the field, and, in particular, if the characteristic of the field is at least $ 5 $ , then such carpets exist for every type. The results obtained imply the existence of irreducible closed elementary matrix carpets over a field of arbitrary characteristic which cannot be extended to full matrix carpets. Examples of such elementary matrix carpets over fields of characteristics $ 2 $ and $ 0 $ were indicated earlier by Koibaev [Trudy Inst. Mat. i Mekh. UrO RAN, 2011; Sib. Math. J., 2021].