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We introduce the notion of an (α, β, γ) triple system, which generalizes the familiar generalized Jordan triple system related to a construction of simple Lie algebras. We then discuss its realization by considering some bilinear algebras and vice versa. Next, as a new concept, we study triality relations (a triality group and a triality derivation) associated with these triple systems; the relations are a generalization of the automorphisms and derivations of the triple systems. Also, we provide examples of several involutive triple systems with a tripotent element.

For a symmetrizable Kac–Moody Lie algebra g , we construct a family of weighted quivers Q m (g) ( m ≥ 2 ) whose cluster modular group Γ Q m (g) contains the Weyl group W (g) as a subgroup. We compute explicit formulae for the corresponding cluster A - and X -transformations. As a result, we obtain green sequences and the cluster Donaldson–Thomas transformation for Q m (g) in a systematic way when g is of finite type. Moreover if g is of classical finite type with the Coxeter number h, the quiver Q kh (g) ( k ≥ 1 ) is mutation-equivalent to a quiver encoding the cluster structure of the higher Teichmüller space of a once-punctured disk with 2k marked points on the boundary, up to frozen vertices. This correspondence induces the action of direct products of Weyl groups on the higher Teichmüller space of a general marked surface. We finally prove that this action coincides with the one constructed in Goncharov and Shen (Adv Math 327:225–348, 2018) from the geometrical viewpoint. [ABSTRACT FROM AUTHOR]

We introduce axial representations and modules over axial algebras as new tools to study axial algebras. All known interesting examples of axial algebras fall into this setting, in particular the Griess algebra whose automorphism group is the Monster group. Our results become especially interesting for Matsuo algebras. We vitalize the connection between Matsuo algebras and 3-transposition groups by relating modules over Matsuo algebras with representations of 3-transposition groups. As a by-product, we define, given a Fischer space, a group that can fulfill the role of a universal 3-transposition group. [ABSTRACT FROM AUTHOR]

Let R be any ring, M an R-module and G a subgroup of $$\hbox {Aut}_{\mathrm{R}}\hbox {M}$$ of finite rank. We compare the R-module structure of certain R-G images of M with that of certain R-G submodules of M. (We are treating M here as an R-G bimodule). For example, if A $$=$$ [M, G] and $$\hbox {B}=\hbox {M/C}_{\mathrm{M}}\hbox {(G)}$$ , then we prove that as R-modules, A is Artinian if B is Artinian, B is Noetherian if A is Noetherian and hence A has a composition series of finite length if and only if B does. [ABSTRACT FROM AUTHOR]

Let k be a perfect field of characteristic $$p > 0$$ , and let G be a finite group. We consider the pointed G-curves over k associated by Harbater, Katz, and Gabber to faithful actions of G on k[[ t]] over k. We use such 'HKG G-curves' to classify the automorphisms of k[[ t]] of p-power order that can be expressed by particularly explicit formulas, namely those mapping t to a power series lying in a $$\mathbb {Z}/p\mathbb {Z}$$ Artin-Schreier extension of k( t). In addition, we give necessary and sufficient criteria to decide when an HKG G-curve with an action of a larger finite group J is also an HKG J-curve. [ABSTRACT FROM AUTHOR]

We investigate the recently obtained condition [] for amenability of groups generated by bounded automorphisms of a spherically homogeneous rooted tree together with our group construction [] based on the notion of a homogeneous automorphism and its crack. We establish the general condition answering the question when a group generated by homogeneous automorphisms and their cracks is amenable. We apply this result to the universal construction from [], providing finitely generated, amenable groups, which are dense in infinitely iterated, permutational wreath products of transitive, Abelian groups. [ABSTRACT FROM AUTHOR]

Given a cardinal $\lambda $ with $\lambda =\lambda ^{\aleph _0}$ , we show that there is a field of cardinality $\lambda $ whose automorphism group is a free group of rank $2^\lambda $ . In the proof of this statement, we develop general techniques that enable us to realize certain groups as the automorphism group of structures of a given cardinality. They allow us to show that analogues of this result hold for free objects in various varieties of groups. For example, the free abelian group of rank $2^\lambda $ is the automorphism group of a field of cardinality $\lambda $ whenever $\lambda $ is a cardinal with $\lambda =\lambda ^{\aleph _0}$ . Moreover, we apply these techniques to show that consistently the assumption that $\lambda =\lambda ^{\aleph _0}$ is not necessary for the existence of a field of cardinality $\lambda $ whose automorphism group is a free group of rank $2^\lambda $ . Finally, we use them to prove that the existence of a cardinal $\lambda $ of uncountable cofinality with the property that there is no field of cardinality $\lambda $ whose automorphism group is a free group of rank greater than $\lambda $ implies the existence of large cardinals in certain inner models of set theory.

Let $$(H_i)_{i\ge 1}$$ be an arbitrary sequence of non-Abelian finite simple transitive permutation groups. By using the combinatorial language of time-varying automata, we provide an explicit and naturally defined construction of a two-element set which generates a dense subgroup in the inverse limit $$\ldots \wr H_2\wr H_1$$ of iterated permutational wreath products of the groups $$H_i$$ . The corresponding automaton is equipped with three states, one of which is neutral and the semigroup generated by the other two states is free. We derive other algebraic and geometric properties of the group generated by this automaton. By using the notion of a Mealy automaton, we obtain the analogous construction for the infinite permutational wreath power of an arbitrary non-Abelian finite simple transitive permutation group $$H$$ on a set $$X$$ . We show that the wreath power $$\ldots \wr H\wr H$$ contains a dense 2-generated not finitely presented amenable subgroup of exponential growth, which is generated by a 3-state Mealy automaton over the alphabet $$X$$ . The self-similar group generated by this automaton is self-replicating, contracting and regular weakly branch over the commutator subgroup. [ABSTRACT FROM AUTHOR]

We use the combinatorial language of automata to define and study profinite groups which are infinitely iterated permutational wreath products of transitive finite permutation groups. We provide some naturally defined automaton realizations of these groups by the so-called time-varying automata, as well as by Mealy automata to characterize these groups as automaton groups, i.e. as groups topologically generated by a single automaton. [ABSTRACT FROM AUTHOR]

Let F be a field, A be a vector space over F, and GL( F,A) the group of all automorphisms of the vector space A. A subspace B of A is called nearly G-invariant, if dim F ( BFG/B) is finite. A subspace B is called almost G-invariant, if dim F ( B/Core G ( B)) is finite. In the present article we begin the study of subgroups G of GL( F,A) such that every subspace of A is either nearly G-invariant or almost G-invariant. More precisely, we consider the case when G is a periodic p′-group where p = char F. [ABSTRACT FROM AUTHOR]