On Automorphisms of the Affine Cremona Group

Hanspeter Kraft; Immanuel Stampfli

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Coleman automorphisms of finite groups with a self-centralizing normal subgroup

Jinke Hai (2020)

Czechoslovak Mathematical Journal

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Let $G$ be a finite group with a normal subgroup $N$ such that $C_{G} (N) \leq N$ . It is shown that under some conditions, Coleman automorphisms of $G$ are inner. Interest in such automorphisms arose from the study of the normalizer problem for integral group rings.

The tame automorphism group of an affine quadric threefold acting on a square complex

Cinzia Bisi, Jean-Philippe Furter, Stéphane Lamy (2014)

Journal de l’École polytechnique — Mathématiques

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We study the group $Tame ({SL}_{2})$ of tame automorphisms of a smooth affine $3$ -dimensional quadric, which we can view as the underlying variety of ${SL}_{2} (ℂ)$ . We construct a square complex on which the group admits a natural cocompact action, and we prove that the complex is $CAT (0)$ and hyperbolic. We propose two applications of this construction: We show that any finite subgroup in $Tame ({SL}_{2})$ is linearizable, and that $Tame ({SL}_{2})$ satisfies the Tits alternative.

Automorphisms of $(λ) / ℐ_{κ}$

Paul Larson, Paul McKenney (2016)

Fundamenta Mathematicae

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We study conditions on automorphisms of Boolean algebras of the form $(λ) / ℐ_{κ}$ (where λ is an uncountable cardinal and $ℐ_{κ}$ is the ideal of sets of cardinality less than κ ) which allow one to conclude that a given automorphism is trivial. We show (among other things) that every automorphism of $(2^{κ}) / ℐ_{κ ⁺}$ which is trivial on all sets of cardinality κ⁺ is trivial, and that $M A_{ℵ ₁}$ implies both that every automorphism of (ℝ)/Fin is trivial on a cocountable set and that every automorphism of (ℝ)/Ctble is trivial. ...

Artinian automorphisms of infinite groups

Antonella Leone (2006)

Bollettino dell'Unione Matematica Italiana

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An automorphism a of a group G is called an artinian automorphism if for every strictly descending chain $H_{1}>H_{2}>\cdots>H_{n}>\cdots$ of subgroups of G there exists a positive integer $m$ such that $(H_{n})^{a}=H_{n}$ for every $n\geq m$ . In this paper we show that in many cases the group of all artinian automorphisms of $G$ coincides with the group of all power automorphisms of $G$ .

Infinitesimal CR automorphisms for a class of polynomial models

Martin Kolář, Francine Meylan (2017)

Archivum Mathematicum

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In this paper we study infinitesimal CR automorphisms of Levi degenerate hypersurfaces. We illustrate the recent general results of [18], [17], [15], on a class of concrete examples, polynomial models in $ℂ^{3}$ of the form $ℑ w = ℜ (P (z) \bar{Q (z)})$ , where $P$ and $Q$ are weighted homogeneous holomorphic polynomials in $z = (z_{1}, z_{2})$ . We classify such models according to their Lie algebra of infinitesimal CR automorphisms. We also give the first example of a non monomial model which admits a nonlinear rigid automorphism. ...

Automorphisms and derivations of a Fréchet algebra of locally integrable functions

F. Ghahramani, J. McClure (1992)

Studia Mathematica

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We find representations for the automorphisms, derivations and multipliers of the Fréchet algebra $L ¹_{l o c}$ of locally integrable functions on the half-line $ℝ^{+}$ . We show, among other things, that every automorphism θ of $L ¹_{l o c}$ is of the form $θ = φ_{a} e^{λ X} e^{D}$ , where D is a derivation, X is the operator of multiplication by coordinate, λ is a complex number, a > 0, and $φ_{a}$ is the dilation operator $(φ_{a} f) (x) = a f (a x)$ ( $f \in L ¹_{l o c}$ , $x \in ℝ^{+}$ ). It is also shown that the automorphism group is a topological group with the topology of uniform convergence...

The group of automorphisms of $L_{\infty}$ is algebraically reflexive

Félix Cabello Sánchez (2004)

Studia Mathematica

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We study the reflexivity of the automorphism (and the isometry) group of the Banach algebras $L_{\infty} (μ)$ for various measures μ. We prove that if μ is a non-atomic σ-finite measure, then the automorphism group (or the isometry group) of $L_{\infty} (μ)$ is [algebraically] reflexive if and only if $L_{\infty} (μ)$ is *-isomorphic to $L_{\infty} [0, 1]$ . For purely atomic measures, we show that the group of automorphisms (or isometries) of $ℓ_{\infty} (Γ)$ is reflexive if and only if Γ has non-measurable cardinal. So, for most “practical” purposes, the automorphism...

On automorphisms of the Banach space $ℓ_{\infty} / c ₀$

Piotr Koszmider, Cristóbal Rodríguez-Porras (2016)

Fundamenta Mathematicae

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We investigate Banach space automorphisms $T : ℓ_{\infty} / c ₀ \to ℓ_{\infty} / c ₀$ focusing on the possibility of representing their fragments of the form $T_{B, A} : ℓ_{\infty} (A) / c ₀ (A) \to ℓ_{\infty} (B) / c ₀ (B)$ for A,B ⊆ ℕ infinite by means of linear operators from $ℓ_{\infty} (A)$ into $ℓ_{\infty} (B)$ , infinite A×B-matrices, continuous maps from B* = βB∖B into A*, or bijections from B to A. This leads to the analysis of general bounded linear operators on $ℓ_{\infty} / c ₀$ . We present many examples, introduce and investigate several classes of operators, for some of them we obtain satisfactory representations and for...

Polycyclic groups with automorphisms of order four

Tao Xu, Fang Zhou, Heguo Liu (2016)

Czechoslovak Mathematical Journal

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In this paper, we study the structure of polycyclic groups admitting an automorphism of order four on the basis of Neumann’s result, and prove that if $α$ is an automorphism of order four of a polycyclic group $G$ and the map $ϕ : G \to G$ defined by $g^{ϕ} = [g, α]$ is surjective, then $G$ contains a characteristic subgroup $H$ of finite index such that the second derived subgroup $H^{''}$ is included in the centre of $H$ and $C_{H} (α^{2})$ is abelian, both $C_{G} (α^{2})$ and $G / [G, α^{2}]$ are abelian-by-finite. These results extend recent and classical results in...

Automorphisms of the algebra of operators in $^{p}$ preserving conditioning

Ryszard Jajte (2010)

Colloquium Mathematicae

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Let α be an isometric automorphism of the algebra $_{p}$ of bounded linear operators in $^{p} [0, 1]$ (p ≥ 1). Then α transforms conditional expectations into conditional expectations if and only if α is induced by a measure preserving isomorphism of [0, 1].

The Banach-Lie Group of Lie Automorphisms of an $H^{*}$ -Algebra

Antonio J. Calderón Martín, Candido Martín González (2007)

Bollettino dell'Unione Matematica Italiana

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We study the Banach-Lie group $\operatorname{Aut}(A^{-})$ of Lie automorphisms of a complex associative $H^{*}$ -algebra. Also some consequences about its Lie algebra, the algebra of Lie derivations of $A$ , are obtained. For a topologically simple $A$ , in the infinite-dimensional case we have $\operatorname{Aut}(A^{-})_{0}=\operatorname{Aut}(A)$ implying $\operatorname{Der}(A)=\operatorname{Der}(A^{-})$ . In the finite dimensional case $\operatorname{Aut}(A^{-})_{0}$ is a direct product of $\operatorname{Aut}(A)$ and a certain subgroup of Lie derivations $\delta$ from $A$ to its center, annihilating commutators.

Equidistribution towards the Green current for holomorphic maps

Tien-Cuong Dinh, Nessim Sibony (2008)

Annales scientifiques de l'École Normale Supérieure

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Let $f$ be a non-invertible holomorphic endomorphism of a projective space and $f^{n}$ its iterate of order $n$ . We prove that the pull-back by $f^{n}$ of a generic (in the Zariski sense) hypersurface, properly normalized, converges to the Green current associated to $f$ when $n$ tends to infinity. We also give an analogous result for the pull-back of positive closed $(1, 1)$ -currents and a similar result for regular polynomial automorphisms of $ℂ^{k}$ .

Automorphism group of green algebra of weak Hopf algebra corresponding to Sweedler Hopf algebra

Liufeng Cao, Dong Su, Hua Yao (2023)

Czechoslovak Mathematical Journal

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Let $r (𝔴_{2}^{0})$ be the Green ring of the weak Hopf algebra $𝔴_{2}^{0}$ corresponding to Sweedler’s 4-dimensional Hopf algebra $H_{2}$ , and let $Aut (R (𝔴_{2}^{0}))$ be the automorphism group of the Green algebra $R (𝔴_{2}^{0}) = r (𝔴_{2}^{0}) \otimes_{ℤ} ℂ$ . We show that the quotient group $Aut (R (𝔴_{2}^{0})) / C_{2} ≅ S_{3}$ , where $C_{2}$ contains the identity map and is isomorphic to the infinite group $(ℂ^{*}, \times)$ and $S_{3}$ is the symmetric group of order 6.

On strongly affine extensions of commutative rings

Nabil Zeidi (2020)

Czechoslovak Mathematical Journal

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A ring extension $R \subseteq S$ is said to be strongly affine if each $R$ -subalgebra of $S$ is a finite-type $R$ -algebra. In this paper, several characterizations of strongly affine extensions are given. For instance, we establish that if $R$ is a quasi-local ring of finite dimension, then $R \subseteq S$ is integrally closed and strongly affine if and only if $R \subseteq S$ is a Prüfer extension (i.e. $(R, S)$ is a normal pair). As a consequence, the equivalence of strongly affine extensions, quasi-Prüfer extensions and INC-pairs is shown....

Relative exactness modulo a polynomial map and algebraic $(ℂ^{p}, +)$ -actions

Philippe Bonnet (2003)

Bulletin de la Société Mathématique de France

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Let $F = (f_{1}, ..., f_{q})$ be a polynomial dominating map from $ℂ^{n}$ to $ℂ^{q}$ . We study the quotient $𝒯^{1} (F)$ of polynomial 1-forms that are exact along the generic fibres of $F$ , by 1-forms of type $d R + \sum a_{i} d f_{i}$ , where $R, a_{1}, ..., a_{q}$ are polynomials. We prove that $𝒯^{1} (F)$ is always a torsion $ℂ [t_{1}, ..., t_{q}]$ -module. Then we determine under which conditions on $F$ we have $𝒯^{1} (F) = 0$ . As an application, we study the behaviour of a class of algebraic $(ℂ^{p}, +)$ -actions on $ℂ^{n}$ , and determine in particular when these actions are trivial.

Homology of origamis with symmetries

Carlos Matheus, Jean-Christophe Yoccoz, David Zmiaikou (2014)

Annales de l’institut Fourier

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Given an origami (square-tiled surface) $M$ with automorphism group $Γ$ , we compute the decomposition of the first homology group of $M$ into isotypic $Γ$ -submodules. Through the action of the affine group of $M$ on the homology group, we deduce some consequences for the multiplicities of the Lyapunov exponents of the Kontsevich-Zorich cocycle. We also construct and study several families of interesting origamis illustrating our results.

On automorphisms fixing subnormal subgroups of soluble groups

Silvana Franciosi, Francesco de Giovanni (1988)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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The group $Aut_{sn}G$ of all automorphisms leaving invariant every subnormal subgroup of the group $G$ is studied. In particular it is proved that $Aut_{sn}G$ is metabelian if $G$ is soluble, and that $Aut_{sn}G$ is either finite or abelian if $G$ is polycyclic.

Automorphisms of metacyclic groups

Haimiao Chen, Yueshan Xiong, Zhongjian Zhu (2018)

Czechoslovak Mathematical Journal

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A metacyclic group $H$ can be presented as $〈 α, β : α^{n} = 1$ , $β^{m} = α^{t}$ , $β α β^{- 1} = α^{r} 〉$ for some $n$ , $m$ , $t$ , $r$ . Each endomorphism $σ$ of $H$ is determined by $σ (α) = α^{x_{1}} β^{y_{1}}$ , $σ (β) = α^{x_{2}} β^{y_{2}}$ for some integers $x_{1}$ , $x_{2}$ , $y_{1}$ , $y_{2}$ . We give sufficient and necessary conditions on $x_{1}$ , $x_{2}$ , $y_{1}$ , $y_{2}$ for $σ$ to be an automorphism.

Automorphisms and generalized skew derivations which are strong commutativity preserving on polynomials in prime and semiprime rings

Vincenzo de Filippis (2016)

Czechoslovak Mathematical Journal

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Let $R$ be a prime ring of characteristic different from 2, $Q_{r}$ its right Martindale quotient ring and $C$ its extended centroid. Suppose that $F$ , $G$ are generalized skew derivations of $R$ with the same associated automorphism $α$ , and $p (x_{1}, ..., x_{n})$ is a non-central polynomial over $C$ such that $[F (x), α (y)] = G ([x, y])$ for all $x, y \in {p (r_{1}, ..., r_{n}) : r_{1}, ..., r_{n} \in R}$ . Then there exists $λ \in C$ such that $F (x) = G (x) = λ α (x)$ for all $x \in R$ .