On Regular Subgroups of the Affine Group

Abstract

Catino and Rizzo ['Regular subgroups of the affine group and radical circle algebras', Bull. Aust. Math. Soc. 79 (2009), 103-107] established a link between regular subgroups of the affine group and the radical brace over a field on the underlying vector space. We propose new constructions of radical braces that allow us to obtain systematic constructions of regular subgroups of the affine group. In particular, this approach allows to put in a more general context the regular subgroups constructed in Tamburini Bellani ['Some remarks on regular subgroups of the affine group' Int.

More on regular subgroups of the affine group

“…So δ(v 2 ) = 0 0 0 α2 0 0 β2 1 0 and δ(v 3 ) = 0 0 0 α3 γ1 γ2 β3 γ3 −γ1 . Applying (2) to v 2 , v 2 and to v 3 , v 2 we obtain respectively α 2 = 0 and γ 1 = −1, γ 2 = 0, which contradicts the unipotency of U . On the other hand, if x 3 = 0, conjugating by diag I 2 ,…”

“…In particular, there is a bijection between conjugacy classes of abelian regular subgroups of AGL n (F) and isomorphism classes of abelian split local algebras of dimension n + 1 over F. This fact was first observed in [1]. It was studied also in connection with other algebraic structures in [3,4,5,6] and in [2], where the classification of nilpotent associative algebras given in [7] is relevant.…”

The matched product of set-theoretical solutions of the Yang-Baxter equation

“…Such solutions have been relatively investigated [9,12,16,28,32]. Further advancements in the field of skew braces relating to Hopf-Galois structures can be found in [12,28,32], whereas [9], an extension of [4], partially answered the extension problem in a simplified case. Lebed in [23] drew attention on idempotent solutions that, although of little interest in physics, provide a tool for dealing with very different algebraic structures ranging from free (commutative) monoids to factorizable monoids, and from distributive lattices to Young tableaux and plactic monoids.…”

The Matched Product of the Solutions to the Yang–Baxter Equation of Finite Order

“…A set B with two operations + and • is a skew left brace if (B, +) and (B, •) are groups and the condition a • (b + c) = a • b − a + a • c is satisfied for all a, b, c ∈ B. Further advancements in the field of skew braces relating to Hopf-Galois structures can be found in [32,14,27], whereas [10], an extension of [7], partially answered the extension problem in a simplified case. It is worth mentioning that in literature bijective solutions are usually defined on finite sets.…”