Related papers: The Nagata automorphism is shifted linearizable

Let $\mathcal{L}$ be a finite-dimensional semisimple Lie algebra of rank $N$ over an algebraically closed field of characteristic $0$. Associated to $\mathcal{L}$ is a family of polynomial folding maps…

We study automorphisms of the free associative algebra K<x,y,z> over a field K which fix z and such that the images of x, y are linear with respect to x, y. We prove that some of these automorphisms are wild in the class of all…

Automorphic loops are loops in which all inner mappings are automorphisms. A large class of automorphic loops is obtained as follows: Let $m$ be a positive even integer, $G$ an abelian group, and $\alpha$ an automorphism of $G$ that…

We consider the family $f_{a,b}(x,y)=(y,(y+a)/(x+b))$ of birational maps of the plane and the parameter values $(a,b)$ for which $f_{a,b}$ gives an automorphism of a rational surface. In particular, we find values for which $f_{a,b}$ is an…

Let $K$ be a field, and let $\Aut \,K^2$ be the group of polynomial automorphisms of $K^2$. We investigate which subgroups are linear or not. In characteristic zero, there are small nonlinear subgroups and some big linear subgroups. When…

Let ${\mathfrak g}$ be a finite dimensional simple Lie algebra over an algebraically closed field $K$ of characteristic $0$. A linear map $\varphi:{\mathfrak g}\to {\mathfrak g}$ is called a local automorphism if for every $x$ in…

In this paper we consider some classical varieties of linear algebras over the field which has characteristic 0. For every considered variety we take a category of the finite generated free algebras of this variety. And for every this…

An algebra $L$ over a field $\Bbb F$, in which product is denoted by $[\,,\,]$, is said to be \textit{ Lie type algebra} if for all elements $a,b,c\in L$ there exist $\alpha, \beta\in \Bbb F$ such that $\alpha\neq 0$ and $[[a,b],c]=\alpha…

In 2004 Shestakov and Umirbaev proved that the Nagata automorphism of the polynomial algebra in three variables is wild. We fix a Z-grading on this algebra and consider graded-wild automorphisms, i.e. such automorphisms that can not be…

For a finite alphabet $\mathcal{A}$ and shift $X\subseteq\mathcal{A}^{\mathbb{Z}}$ whose factor complexity function grows at most linearly, we study the algebraic properties of the automorphism group ${\rm Aut}(X)$. For such systems, we…

Automorphisms of algebras $R$ from a very large axiomatic class of quantum nilpotent algebras are studied using techniques from noncommutative unique factorization domains and quantum cluster algebras. First, the Nakayama automorphism of…

Let $k$ be a field of characteristic $p>0$. We discuss the automorphisms of the polynomial ring $k[x_1,\ldots ,x_n]$ of order $p$, or equivalently the ${\bf Z}/p{\bf Z}$-actions on the affine space ${\bf A}_k^n$. When $n=2$, such an…

We study the actions of a Lie group $G$ by birationally extendible automorphisms on a domain $D\subset C^n$. For a large class of such domains defined by polynomial inequalities, all automorphisms are of this type. In the cases 1) $G$ has…

Let $G_{g,b}$ be the set of all uni/trivalent graphs representing the combinatorial structures of pant decompositions of the oriented surface of genus $g$ with $b$ boundary components. We describe the set $A_{g,b}$ of all automorphisms of…

A polynomial automorphism of $\mathbb{A}^n$ over a field of characteristic zero is called co-tame if, together with the affine subgroup, it generates the entire tame subgroup. We prove some new classes of automorphisms, including…

Let $\nabla$ be a meromorphic connection on a vector bundle over a compact Riemann surface $\Gamma$. An automorphism $\sigma:\Gamma\to\Gamma$ is called a symmetry of $\nabla$ if the pull-back bundle and the pull-back connection can be…

Consider a normal projective variety $X$, a linear algebraic subgroup $G$ of Aut($X$), and the field $K$ of $G$-invariant rational functions on $X$. We show that the subgroup of Aut($X$) that fixes $K$ pointwise is linear algebraic. If $K$…

We prove that every 2-local automorphism of the unitary group or the general linear group on a complex infinite-dimensional separable Hilbert space is an automorphism. Thus these types of transformations are completely determined by their…

Let $A$ be a Koszul Artin-Schelter regular algebra and $\sigma$ an algebra homomorphism from $A$ to $M_{2\times 2}(A)$. We compute the Nakayama automorphisms of a trimmed double Ore extension $A_P[y_1, y_2; \sigma]$ (introduced in…

Planar polynomial automorphisms are polynomial maps of the plane whose inverse is also a polynomial map. A map is reversible if it is conjugate to its inverse. Here we obtain a normal form for automorphisms that are reversible by an…