Related papers: Automorphisms fixing a variable of K<x,y,z>

The group of 2-by-2 matrices with integer entries and determinant $\pm > 1$ can be identified either with the group of outer automorphisms of a rank two free group or with the group of isotopy classes of homeomorphisms of a 2-dimensional…

In one of our recent papers, the associative and the Lie algebras of Weyl type $A[D]=A\otimes F[D]$ were defined and studied, where $A$ is a commutative associative algebra with an identity element over a field $F$ of any characteristic,…

We construct an analogue of the Nagata automorphism of free nonassociative algebras and free commutative algebras of rank two over a Euclidean domain and prove that it is wild.

Let $A_1,...,A_k$ be a system of free factors of $F_n$. The group of relative automorphisms $Aut(F_n;A_1,...,A_k)$ is the group given by the automorphisms of $F_n$ that restricted to each $A_i$ are conjugations by elements in $F_n$. The…

Let $G$ be a connected graph on $n$ vertices and $1 \le k \le n-1$ an integer. The $k$-token graph of $G$ is the graph $F_k(G)$ whose vertices are all the $k$-subsets of vertices of $G$, two of which are adjacent whenever their symmetric…

We study automorphism groups of formal matrix algebras. We also consider automorphisms of ordinary matrix algebras (in particular, triangular matrix algebras).

This note is devoted, after the result of Harui, arXiv:1306.5842, to solve some natural questions for non-singular plane curves of degree $d$ over an algebraically closed field $K$ of zero characteristic.

We construct k-parameter families of rational surface automorphisms for any k. These are automorphisms of surfaces X, which are constructed from iterated blowups over the projective plane. In certain cases: we are able to determine the…

We prove that if a field k is infinite, char(k)=0 and k has not nontrivial automorphisms then automorphic equivalence of representations of Lie algebras coincide with geometric equivalence. We achieve our result by consideration of 1-sorted…

Let GF(q)[x,y] be the polynomial algebra in two variables over the finite field GF(q) with q elements. We give an exact formula and the asymptotics for the number p(n) of automorphisms (f,g) of GF(q)[x,y] such that max{deg(f),deg(g)}=n. We…

In a recent paper [3], the authors introduced a map $\mathcal{F}$ which associates a Deitmar scheme (which is defined over the field with one element, denoted by $\mathbb{F}_1$) with any given graph $\Gamma$. By base extension, a scheme…

In this paper we prove that a pure, regular, totally odd, polarizable weakly compatible system of $l$-adic representations is potentially automorphic. The innovation is that we make no irreducibility assumption, but we make a purity…

Let $F_n$ be the free group on $n\ge 2$ elements and $\A(F_n)$ its group of automorphisms. In this paper we present a rich collection of linear representations of $\A(F_n)$ arising through the action of finite index subgroups of it on…

We apply the theory of localization for tame and wild coalgebras in order to prove the following theorem: "Let Q be an acyclic quiver. Then any tame admissible subcoalgebra of KQ is the path coalgebra of a quiver with relations".

We investigate the size of fixed point sets of automorphisms of bounded domains in $\mathbb{C}^n$. In one complex variable, a nontrivial automorphism has at most two fixed points, but in higher dimensions fixed point sets need not be…

In this paper, we determine the automorphism groups of generalized Kausz compactifications $\mathcal T_{s,p,n}$. By establishing the (semi-)positivity of the anticanonical bundles of $\mathcal T_{s,p,n}$, we also determine the automorphism…

Let $K$ be an {\em arbitrary} field of characteristic $p>0$, let $A$ be one of the following algebras: $P_n:= K[x_1, ..., x_n]$ is a polynomial algebra, $\CD (P_n)$ is the ring of differential operators on $P_n$, $\CD (P_n)\t P_m$, the…

The groups of automorphisms of the Lie algebras of formally analytic vector fields with constant divergence are found.

We show that every automorphism $\alpha$ of a free group $F_k$ of finite rank $k$ has {\it asymptotically periodic} dynamics on $F_k$ and its boundary $\partial F_k$: there exists a positive power $\alpha^q$ such that every element of the…

We prove that every $2$-local automorphism on a finite-dimensional semi-simple Lie algebra $\mathcal{L}$ over an algebraically closed field of characteristic zero is an automorphism. We also show that each finite-dimensional nilpotent Lie…