Minimality in topological groups and Heisenberg type groups

We study relatively minimal subgroups in topological groups. We find, in particular, some natural relatively minimal subgroups in unipotent groups which are defined over “good” rings. By “good” rings we mean archimedean absolute valued (not necessarily associative) division rings. Some of the classical rings which we consider besides the field of reals are the ring of quaternions and the ring of octonions. This way we generalize in part a previous result which was obtained by Dikranjan and Megrelishvili and involved the Heisenberg group.

💡 Research Summary

The paper investigates relatively minimal and co‑minimal subgroups within topological groups, focusing on generalized Heisenberg groups built from biadditive maps. After recalling the notions of minimality (a group admitting no strictly coarser Hausdorff topology), relative minimality (a subset whose induced topology cannot be weakened), and co‑minimality (the quotient topology is forced to be the original one), the author extends earlier results that were limited to real‑valued settings.

The central technical achievement is Theorem 3.4, which shows that for any division ring F equipped with an Archimedean absolute value (the ring may be non‑associative, e.g., quaternions ℍ or octonions 𝔾), the bilinear form
 wₙ : Fⁿ × Fⁿ → F, wₙ( x̄,ȳ ) = ∑_{i=1}^{n} x_i y_i
is strongly minimal. In other words, no strictly coarser Hausdorff group topologies on the domain can keep wₙ continuous. The proof uses the metric induced by the absolute value and a contradiction argument based on unbounded neighborhoods in a hypothetically coarser topology.

From this strong minimality, Theorem 2.9 (a known result) yields that in the generalized Heisenberg group H(wₙ) = (F × Fⁿ) ⋉ Fⁿ, the subgroups F × {0}, (F × Fⁿ) ⋉ {0}, and (F × {0}) ⋉ Fⁿ are co‑minimal, while ({0} × Fⁿ) ⋉ Fⁿ and its obvious variants are relatively minimal. Corollary 3.6 lists these subgroups explicitly.

The paper then turns to upper unitriangular matrix groups U_{n+2}(F). Theorem 3.9 proves that for any indices i < j with (i,j) ≠ (1,n+2), the one‑parameter subgroup consisting of matrices that are identity except for a single entry a_{ij} in position (i,j) is relatively minimal in U_{n+2}(F). The argument relies on embedding a relatively minimal Heisenberg‑type subset into U_{n+2}(F) and applying Lemma 3.8, which shows that relative minimality is preserved under subgroup inclusion and topological isomorphisms.

Examples demonstrate the applicability to ℚ, ℝ, ℂ, the quaternion algebra, and the octonion algebra. The results thus generalize earlier work of Dikranjan and Megrelishvili, which dealt only with real Heisenberg groups, to a broad class of non‑commutative and even non‑associative division rings.

In summary, the article establishes that strongly minimal biadditive maps exist over any Archimedean valued division ring, and that these maps generate a rich family of relatively minimal and co‑minimal subgroups in both generalized Heisenberg groups and higher‑dimensional unitriangular groups. This advances the theory of minimal topological groups, opens new avenues for studying non‑commutative topological algebra, and suggests further exploration into more exotic valued rings and their associated minimality phenomena.