Groups of triangular automorphisms of a free associative algebra and a polynomial algebra

Automorphism groups of superextensions of finite monogenic semigroups

A family $\mathcal L$ of subsets of a set $X$ is called linked if $A\cap B\ne\emptyset$ for any $A,B\in\mathcal L$. A linked family $\mathcal M$ of subsets of $X$ is maximal linked if $\mathcal M$ coincides with each linked family $\mathcal L$ on $X$ that contains $\mathcal M$. The superextension $λ(X)$ of $X$ consists of all maximal linked families on $X$. Any associative binary operation $* : X\times X \to X$ can be extended to an associative binary operation $*: λ(X)\timesλ(X)\toλ(X)$. In the paper we study automorphisms of the superextensions of finite monogenic semigroups and characterist…

On the Andreadakis equality for a subgroup of the McCool group

The McCool group has families of subgroups such as the ordinary pure braid group, the upper triangular McCool group and the partial inner automorphism group. The generalized Andreadakis conjecture holds for the ordinary pure braid group and the upper triangular McCool group. In this paper, we prove a similar result for the partial inner automorphism group and also establish an isomorphism between this group and the inner automorphism group of the upper triangular McCool group.

轴向代数的自同构群

Automorphism groups of axial algebras

本文针对轴向代数中因维数重复出现而可能存在的意外同构问题，提出计算完全自同构群的方法，并在最高151维的若干例子上进行应用验证。

关于交换代数自同构群可解性的研究

On Solvability of Automorphism Groups of Commutative Algebras

本文研究了交换代数自同构群的可解性，给出了新的充分条件以保证其可解性。

正交与有向 Fano 平面、$K_7$ 的三角嵌入以及 Frobenius 群 $F_{21}$ 的几何表示

Orthogonal and oriented Fano planes, triangular embeddings of $K_7,$ and geometrical representations of the Frobenius group $F_{21}$

本文研究21阶Frobenius群F21的几何表示，证明两个正交Fano平面的公共自同构群与适当定向Fano平面的自同构群均同构于F21，并证明K7的任意三角嵌入同构于经典环面双嵌入且面2-可着色，从而给出F21的三种几何表示。

关于若干类非交换双幺半群

On some classes of noncommutative dimonoids

本文研究双幺半群，构造非交换类中的阿贝尔、非阿贝尔及矩形双幺半群，并分析其结构、自同构群和光环。

下三角Nielsen自同构群的下中心列

The Lower Central Series of Right Lower-Triangular Nielsen Automorphism Groups

本文确定秩$n\geq3$自由群的下三角Nielsen自同构群$D_n$的下中心列，推导其滤链、半直积分解及商群基，证明$D_n$是Magnus群。

简单马尔采夫代数布尔幂自同构群中的丰富泛型

Ample generics in automorphism groups of Boolean powers of simple Mal'cev algebras

研究有限简单马尔采夫代数布尔幂自同构群的丰富泛型，通过分阿贝尔和非阿贝尔情形证明，利用模表示、\(n\) - 系统理论及群分解等，还得到\(\mathrm{Homeo}\, 2^\omega\)中逐点稳定子有丰富泛型的结果。

$G_2$ 和 $F_4$ 型非结合 Frobenius 代数

Non-associative Frobenius algebras of type $G_2$ and $F_4$

本文用八元代数和 Albert 代数分别给出了 $G_2$ 和 $F_4$ 型群的交换非结合代数的显式描述，并证明了 $G_2$ 型代数的自同构群等于 $G_2$ 型群本身。

Triangular polynomial maps in characteristic $p$

This paper came to existence out of the desire to understand iterations of strictly triangular polynomial maps over finite fields. This resulted in two connected results: First, we give a generalization of $\F_p$-actions on $\F_p^n$ and their description by "locally iterative higher derivations", namely $\Z$-actions on $\F_p^n$ and show how to describe them by what we call "$\Z$-flows". We prove equivalence between locally finite polynomial automorphisms (LFPEs) over finite fields and $\Z$-flows over finite fields. We elaborate on $\Z$-flows of strictly triangular polynomial maps. Second, we d…

Groups acting on tensor products

Groups preserving a distributive product are encountered often in algebra. Examples include automorphism groups of associative and nonassociative rings, classical groups, and automorphism groups of p-groups. While the great variety of such products precludes any realistic hope of describing the general structure of the groups that preserve them, it is reasonable to expect that insight may be gained from an examination of the universal distributive products: tensor products. We give a detailed description of the groups preserving tensor products over semisimple and semiprimary rings, and presen…

Self-similar groups, automatic sequences, and unitriangular representations

We study natural linear representations of self-similar groups over finite fields. In particular, we show that if the group is generated by a finite automaton, then obtained matrices are automatic. This shows a new relation between two separate notions of automaticity: groups generated by automata and automatic sequences. We also show that if the group acts on the tree by $p$-adic automorphisms, then the corresponding linear representation is a representation by infinite triangular matrices. We relate this observation with the notion of height of an automorphism of a rooted tree due to L.Kalou…

On the Andreadakis problem for subgroups of $IA\_n$

Let $F\_n$ be the free group on $n$ generators. Consider the group $IA\_n$ of automorphisms of $F\_n$ acting trivially on its abelianization. There are two canonical filtrations on $IA\_n$: the first one is its lower central series $Γ\_*$; the second one is the Andreadakis filtration $\mathcal A\_*$, defined from the action on $F\_n$. The Andreadakis problem consists in understanding the difference between these filtrations. Here, we show that they coincide when restricted to the subgroup of triangular automorphisms, and to the pure braid group.

The Farey Maps Modulo N

The Farey map is the universal triangular map whose automorphism group is the classical modular group. We study the quotients of the Farey map by the principal congruence subgroups of the modular group. We also study the structure of the underlying graphs of these quotients.