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Journal articles on the topic 'Automorphisms'

Author: Grafiati

Published: 4 June 2021

Last updated: 1 August 2024

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1

Clemens, John D. "Classifying Borel automorphisms." Journal of Symbolic Logic 72, no. 4 (December 2007): 1081–92. http://dx.doi.org/10.2178/jsl/1203350774.

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§1. Introduction. This paper considers several complexity questions regarding Borel automorphisms of a Polish space. Recall that a Borel automorphism is a bijection of the space with itself whose graph is a Borel set (equivalently, the inverse image of any Borel set is Borel). Since the inverse of a Borel automorphism is another Borel automorphism, as is the composition of two Borel automorphisms, the set of Borel automorphisms of a given Polish space forms a group under the operation of composition. We can also consider the class of automorphisms of all Polish spaces. We will be primarily concerned here with the following notion of equivalence:Definition 1.1. Two Borel automorphisms f and g of the Polish spaces X and Y are said to be Borel isomorphic, f ≅ g, if they are conjugate, i.e. there is a Borel bijection φ: X → Y such that φ ∘ f = g ∘ φ.We restrict ourselves to automorphisms of uncountable Polish spaces, as the Borel automorphisms of a countable space are simply the permutations of the space. Since any two uncountable Polish spaces are Borel isomorphic, any Borel automorphism is Borel isomorphic to some automorphism of a fixed space. Hence, up to Borel isomorphism we can fix a Polish space and represent any Borel automorphism as an automorphism of this space. We will use the Cantor space 2ω (with the product topology) as our representative space.

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2

MASHEVITZKY, G., and B. I. PLOTKIN. "ON AUTOMORPHISMS OF THE ENDOMORPHISM SEMIGROUP OF A FREE UNIVERSAL ALGEBRA." International Journal of Algebra and Computation 17, no. 05n06 (August 2007): 1085–106. http://dx.doi.org/10.1142/s0218196707003974.

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Let U be a universal algebra. An automorphism α of the endomorphism semigroup of U defined by α(φ) = sφs-1 for a bijection s : U → U is called a quasi-inner automorphism. We characterize bijections on U defining such automorphisms. For this purpose, we introduce the notion of a pre-automorphism of U. In the case when U is a free universal algebra, the pre-automorphisms are precisely the well-known weak automorphisms of U. We also provide different characterizations of quasi-inner automorphisms of endomorphism semigroups of free universal algebras and reveal their structure. We apply obtained results for describing the structure of groups of automorphisms of categories of free universal algebras, isomorphisms between semigroups of endomorphisms of free universal algebras, automorphism groups of endomorphism semigroups of free Lie algebras etc.

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3

DUNCAN, BENTON L. "AUTOMORPHISMS OF NONSELFADJOINT DIRECTED GRAPH OPERATOR ALGEBRAS." Journal of the Australian Mathematical Society 87, no. 2 (October 2009): 175–96. http://dx.doi.org/10.1017/s1446788708081007.

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AbstractWe analyze the automorphism group for the norm closed quiver algebras 𝒯+(Q). We begin by focusing on two normal subgroups of the automorphism group which are characterized by their actions on the maximal ideal space of 𝒯+(Q). To further discuss arbitrary automorphisms we factor automorphism through subalgebras for which the automorphism group can be better understood. This allows us to classify a large number of noninner automorphisms. We suggest a candidate for the group of inner automorphisms.

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4

Nurkhaidarov, Ermek. "On Generic Automorphisms." WSEAS TRANSACTIONS ON MATHEMATICS 23 (January 26, 2024): 68–71. http://dx.doi.org/10.37394/23206.2024.23.8.

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In this article we investigate generic automorphisms of countable models. Hodges et al. 1993 introduces the notion of SI (small index) generic automorphisms. They used the existence of small index generics to show the small index property of the model. Truss 1989 defines the notion of Truss generic automorphisms. An automorphism f ofM is called Truss generic if its conjugacy class is comeagre in the automorphism group ofM. We study the relationship between these two types of generic automorphisms. We show that either the countable random graph or a countable arithmetically saturated model of True Arithmetic have both SI generic and Truss generic automorphisms. We prove that the dense linear order has the small index property and Truss-generic automorphisms but it does not have SI generic automorphisms. We also construct an example of a countable structure which has SI generics but it does not have Truss generics.

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Hakuta, Keisuke, and Tsuyoshi Takagi. "Sign of Permutation Induced by Nagata Automorphism over Finite Fields." Journal of Mathematics Research 9, no. 5 (September 7, 2017): 54. http://dx.doi.org/10.5539/jmr.v9n5p54.

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This paper proves that the Nagata automorphism over a finite field can be mimicked by a tame automorphism which is a composition of four elementary automorphisms. By investigating the sign of the permutations induced by the above elementary automorphisms, one can see that if the Nagata automorphism is defined over a prime field of characteristic two, the Nagata automorphism induces an odd permutation, and otherwise, the Nagata automorphism induces an even permutation.

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6

JORDAN, DAVID A., and NONGKHRAN SASOM. "REVERSIBLE SKEW LAURENT POLYNOMIAL RINGS AND DEFORMATIONS OF POISSON AUTOMORPHISMS." Journal of Algebra and Its Applications 08, no. 05 (October 2009): 733–57. http://dx.doi.org/10.1142/s0219498809003564.

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A skew Laurent polynomial ring S = R[x±1;α] is reversible if it has a reversing automorphism, that is, an automorphism θ of period 2 that transposes x and x-1 and restricts to an automorphism γ of R with γ = γ-1. We study invariants for reversing automorphisms and apply our methods to determine the rings of invariants of reversing automorphisms of the two most familiar examples of simple skew Laurent polynomial rings, namely a localization of the enveloping algebra of the two-dimensional non-abelian solvable Lie algebra and the coordinate ring of the quantum torus, both of which are deformations of Poisson algebras over the base field 𝔽. Their reversing automorphisms are deformations of Poisson automorphisms of those Poisson algebras. In each case, the ring of invariants of the Poisson automorphism is the coordinate ring B of a surface in 𝔽3 and the ring of invariants Sθ of the reversing automorphism is a deformation of B and is a factor of a deformation of 𝔽[x1, x2, x3] for a Poisson bracket determined by the appropriate surface.

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7

YAPTI ÖZKURT, Zeynep. "Normal automorphisms of free metabelian Leibniz algebras." Communications Faculty Of Science University of Ankara Series A1Mathematics and Statistics 73, no. 1 (October 9, 2023): 147–52. http://dx.doi.org/10.31801/cfsuasmas.1265768.

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Let $\mathfrak{M}$ be a free metabelian Leibniz algebra generating set $% X=\{x_{1},...,x_{n}\}$ over the field $K$ of characteristic $0$. An automorphism $ \phi $ of $\mathfrak{M}$ is said to be normal automorphism if each ideal of $\mathfrak{M}$ is invariant under $ \phi $. In this work, it is proven that every normal automorphism of $\mathfrak{M}$ is an IA-automorphism and the group of normal automorphisms coincides with the group of inner automorphisms.

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8

Sheng, Yuqiu, Wende Liu, and Yang Liu. "Local Automorphisms and Local Superderivations of Model Filiform Lie Superalgebras." Journal of Mathematics 2024 (March 27, 2024): 1–9. http://dx.doi.org/10.1155/2024/6650997.

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In this paper, we give the forms of local automorphisms (resp. superderivations) of model filiform Lie superalgebra Ln,m in the matrix version. Linear 2-local automorphisms (resp. superderivations) of Ln,m are also characterized. We prove that each linear 2-local automorphism of Ln,m is an automorphism.

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9

Seifizadeh, Parisa, and Amirali Farokhniaee. "The absolute Frattini automorphisms." MATHEMATICA 65 (88), no. 1 (June 15, 2023): 133–38. http://dx.doi.org/10.24193/mathcluj.2023.1.14.

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Let G be a finite non-abelian p-group, where p is a prime number, and Aut(G) be the group of all automorphisms of $G$. An automorphism alpha of $G$ is called absolute central automorphism if, x^{-1}alpha(x) lies in L(G), where L(G) is the absolute center of G. In addition, alpha is an absolute Frattini automorphism if x^{-1}alpha(x) is in Phi(L(G)), where Phi(L(G)) is the Frattini subgroup of the absolute center of G, and let LF(G) denote the group of all such automorphisms of G. Also, we denote by C_{LF(G)}(Z(G)) and C_{LA(G)}(Z(G)), respectively, the group of all absolute Frattini automorphisms and the group of all absolute central automorphisms of G, fixing elementwise the center Z(G) of G . We give necessary and sufficient conditions on a finite non-abelian p-group G of class two such that C_{LF(G)}(Z(G))=C_{LA(G)}(Z(G)). Moreover, we investigate the conditions under which LF(G) is a torsion-free abelian group.

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10

Curran, M. J., and D. J. McCaughan. "Central automorphisms of finite groups." Bulletin of the Australian Mathematical Society 34, no. 2 (October 1986): 191–98. http://dx.doi.org/10.1017/s0004972700010054.

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This paper considers an aspect of the general problem of how the structure of a group influences the structure of its automorphisms group. A recent result of Beisiegel shows that if P is a p-group then the central automorphisms group of P has no normal subgroups of order prime to p. So, roughly speaking, most of the central automorphisms are of p-power order. This generalizes an old result of Hopkins that if Aut P is abelian (so every automorphisms is central), then Aut P is a p-group.This paper uses a different approach to consider the case when P is a π-group. It is shown that the central automorphism group of P has a normal. π′-subgroup only if P has an abelian direct factor whose automorphism group has such a subgroup.

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11

Edo, Eric, and Drew Lewis. "Co-tame polynomial automorphisms." International Journal of Algebra and Computation 29, no. 05 (July 8, 2019): 803–25. http://dx.doi.org/10.1142/s0218196719500292.

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A polynomial automorphism of [Formula: see text] 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 of [Formula: see text], including nonaffine [Formula: see text]-triangular automorphisms, are co-tame. Of particular interest, if [Formula: see text], we show that the statement “Every [Formula: see text]-triangular automorphism is either affine or co-tame” is true if and only if [Formula: see text]; this improves upon positive results of Bodnarchuk (for [Formula: see text], in any dimension [Formula: see text]) and negative results of the authors (for [Formula: see text], [Formula: see text]). The main technical tool we introduce is a class of maps we term translation degenerate automorphisms; we show that all of these are either affine or co-tame, a result that may be of independent interest in the further study of co-tame automorphisms.

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12

ENDIMIONI, GÉRARD. "NORMAL AUTOMORPHISMS OF A FREE METABELIAN NILPOTENT GROUP." Glasgow Mathematical Journal 52, no. 1 (December 4, 2009): 169–77. http://dx.doi.org/10.1017/s0017089509990267.

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AbstractAn automorphism φ of a group G is said to be normal if φ(H) = H for each normal subgroup H of G. These automorphisms form a group containing the group of inner automorphisms. When G is a non-abelian free (or free soluble) group, it is known that these groups of automorphisms coincide, but this is not always true when G is a free metabelian nilpotent group. The aim of this paper is to determine the group of normal automorphisms in this last case.

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13

Malfait, Wim, and Andrzej Szczepański. "Almost-Bieberbach groups with (in)finite outer automorphism group." Glasgow Mathematical Journal 40, no. 1 (March 1998): 47–62. http://dx.doi.org/10.1017/s0017089500032341.

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AbstractIf we investigate symmetry of an infra-nilmanifoldM, the outer automorphism group of its fundamental group (an almost-Bieberbach group) is known to be a crucial object. In this paper, we characterise algebraically almost-Bieberbach groupsEwith finite outer automorphism group Out(E). Inspired by the description of Anosov diffeomorphisms onM, we also present an interesting class of infinite order outer automorphisms. Another possible type of infinite order outer automorphisms arises when comparing Out(E) with the outer automorphism group of the underlying crystallographic group ofE.

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14

Wald, Kevin. "On orbits, of prompt and low computably enumerable sets." Journal of Symbolic Logic 67, no. 2 (June 2002): 649–78. http://dx.doi.org/10.2178/jsl/1190150103.

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AbstractThis paper concerns automorphisms of the computably enumerable sets. We prove two results relating semilow sets and prompt degrees via automorphisms, one of which is complementary to a recent result of Downey and Harrington. We also show that the property of effective simplicity is not invariant under automorphism, and that in fact every promptly simple set is automorphic to an effectively simple set. A major technique used in these proofs is a modification of the Harrington-Soare version of the method of Harrington-Soare and Cholak for constructing Δ30 automorphisms; this modification takes advantage of a recent result of Soare on the extension of “restricted” automorphisms to full automorphisms.

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15

Thaheem, A. B. "On certain decompositional properties of von Neumann algebras." Glasgow Mathematical Journal 29, no. 2 (July 1987): 177–79. http://dx.doi.org/10.1017/s0017089500006819.

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It is well known that if α and β are commuting *-automorphisms of a von Neumann algebra M satisfying the equation α + α-1 = β + β-1 then M can be decomposed into a direct sum of subalgebras Mp and M(l − p) by a central projection p in M such that α = β on Mp and α = β-1 on M(1 − p) (see, for instance, [6], [7], [2]). Originally this equation arose in the Tomita-Takesaki theory (see, for example, [11]) in the form of one-parameter modular automorphism groups and later on it has been studied for arbitrary automorphisms and one-parameter groups of automorphisms on von Neumann algebras [7], [8], [9]. In the case of automorphism groups satisfying the above equation, one has a similar decomposition but this time without assuming the commutativity condition (cf. [7], [8]). For another relevant work on one-parameter groups of automorphisms which is close to our papers [7] and [8], we refer to Ciorănescu and Zsidó [1]. Regarding applications, this equation has been used for arbitrary automorphisms in the geometric interpretation of the Tomita-Takesaki theory [2] and in the case of automorphism groups it has been a fundamental tool in the generalization of the Tomita-Takesaki theory to Jordan algebras [3]. We may remark that the decomposition in the commuting case [6], [7] is much simpler than in the case of automorphism groups in the non-commutative situation [8]. In some cases one can obtain the decomposition for an arbitrary pair of automorphisms without assuming their commutativity but the problem in the general case has been unresolved. Recently we have shown that if α and β are *-automorphisms of a von Neumann algebra M satisfying the equation α + α-1 = β + β-1 (without assuming the commutativity of α and β) then there exists a central projection p in M such that α2= β2 on Mp and α2 = β−2 on M(l − p) [10].

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16

Beattie, Margaret. "Automorphisms of G-Azumaya Algebras." Canadian Journal of Mathematics 37, no. 6 (December 1, 1985): 1047–58. http://dx.doi.org/10.4153/cjm-1985-056-7.

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Let R be a commutative ring, G a finite abelian group of order n and exponent m, and assume n is a unit in R. In [10], F. W. Long defined a generalized Brauer group, BD(R, G), of algebras with a G-action and G-grading, whose elements are equivalence classes of G-Azumaya algebras. In this paper we investigate the automorphisms of a G-Azumaya algebra A and prove that if Picm(R) is trivial, then these automorphisms are all, in some sense, inner.In fact, each of these “inner” automorphisms can be written as the composition of an inner automorphism in the usual sense and a “linear“ automorphism, i.e., an automorphism of the typewith r(σ) a unit in R. We then use these results to show that the group of gradings of the centre of a G-Azumaya algebra A is a direct summand of G, and thus if G is cyclic of order pr, A is the (smash) product of a commutative and a central G-Azumaya algebra.

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17

Adamus, Elżbieta, Paweł Bogdan, Teresa Crespo, and Zbigniew Hajto. "Pascal finite polynomial automorphisms." Journal of Algebra and Its Applications 18, no. 07 (July 2019): 1950124. http://dx.doi.org/10.1142/s021949881950124x.

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The class of Pascal finite polynomial automorphisms is a subclass of the class of locally finite ones allowing a more effective approach. In characteristic zero, a Pascal finite automorphism is the exponential map of a locally nilpotent derivation. However, Pascal finite automorphisms are defined in any characteristic, and therefore constitute a generalization of exponential automorphisms to positive characteristic. In this paper, we prove several properties of Pascal finite automorphisms. We obtain in particular that the Pascal finite property is stable under taking powers but not under composition. This leads us to formulate a generalization of the exponential generators conjecture to arbitrary characteristic.

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18

Thomsen, Klaus. "Automorphisms of homogeneous C*-algebras." Bulletin of the Australian Mathematical Society 33, no. 1 (February 1986): 145–54. http://dx.doi.org/10.1017/s0004972700002975.

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For a homogeneous C*-algebra we identify the quotient of the automorphism group by the locally unitary automorphisms as a subgroup of the homeomorphisms of the spectrum. We sharpen a known criterion on the spectrum that ensures that all locally unitary automorphisms of the algebra are inner.

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Bardakov, Valeriy G., and Mahender Singh. "Extensions and automorphisms of Lie algebras." Journal of Algebra and Its Applications 16, no. 09 (September 9, 2016): 1750162. http://dx.doi.org/10.1142/s0219498817501626.

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Let [Formula: see text] be a short exact sequence of Lie algebras over a field [Formula: see text], where [Formula: see text] is abelian. We show that the obstruction for a pair of automorphisms in [Formula: see text] to be induced by an automorphism in [Formula: see text] lies in the Lie algebra cohomology [Formula: see text]. As a consequence, we obtain a four term exact sequence relating automorphisms, derivations and cohomology of Lie algebras. We also obtain a more explicit necessary and sufficient condition for a pair of automorphisms in [Formula: see text] to be induced by an automorphism in [Formula: see text], where [Formula: see text] is a free nilpotent Lie algebra of rank [Formula: see text] and step [Formula: see text].

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BISWAS, INDRANIL, TOMAS L. GÓMEZ, and VICENTE MUÑOZ. "AUTOMORPHISMS OF MODULI SPACES OF SYMPLECTIC BUNDLES." International Journal of Mathematics 23, no. 05 (May 2012): 1250052. http://dx.doi.org/10.1142/s0129167x12500528.

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Let X be an irreducible smooth complex projective curve of genus g ≥ 4. Fix a line bundle L on X. Let MSp(L) be the moduli space of semistable symplectic bundles (E, φ : E ⊗ E → L) on X, with the symplectic form taking values in L. We show that the automorphism group of MSp(L) is generated by the automorphisms of the form E ↦ E ⊗ M, where [Formula: see text], together with the automorphisms induced by automorphisms of X.

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21

Voiculescu, Dan. "Almost inductive limit automorphisms and embeddings into AF-algebras." Ergodic Theory and Dynamical Systems 6, no. 3 (September 1986): 475–84. http://dx.doi.org/10.1017/s0143385700003618.

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AbstractThe crossed product of an AF-algebra by an automorphism, a power of which is approximately inner, is shown to be embeddable into an AF-algebra. The proof uses almost inductive limit automorphisms, i.e. automorphisms possessing a sequence of almost invariant finite-dimensional C*-subalgebras converging to the given AF-algebra.

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22

BELLINGERI, PAOLO. "ON AUTOMORPHISMS OF SURFACE BRAID GROUPS." Journal of Knot Theory and Its Ramifications 17, no. 01 (January 2008): 1–11. http://dx.doi.org/10.1142/s0218216508005756.

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23

Kutnar, Klavdija, Dragan Marusic, Stefko Miklavic, and Rok Strasek. "Automorphisms of Tabacjn graphs." Filomat 27, no. 7 (2013): 1157–64. http://dx.doi.org/10.2298/fil1307157k.

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A bicirculant is a graph admitting an automorphism whose cyclic decomposition consists of two cycles of equal length. In this paper we consider automorphisms of the so-called Tahacjn graphs, a family of pentavalent bicirculants which are obtained from the generalized Petersen graphs by adding two additional perfect matchings between the two orbits of the above mentioned automorphism. As a corollary, we determine which Tabacjn graphs are vertex-transitive.

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24

Singh, Sandeep, Deepak Gumber, and Hemant Kalra. "IA-automorphisms of finitely generated nilpotent groups." Journal of Algebra and Its Applications 13, no. 07 (May 2, 2014): 1450027. http://dx.doi.org/10.1142/s0219498814500273.

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An automorphism of a group G is called an IA-automorphism if it induces the identity automorphism on the abelianized group G/G′. Let IA (G) denote the group of all IA-automorphisms of G. We classify all finitely generated nilpotent groups G of class 2 for which IA (G) ≃ Inn (G). In particular, we classify all finite nilpotent groups of class 2 for which each IA-automorphism is inner.

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Truss, J. K. "On notions of genericity and mutual genericity." Journal of Symbolic Logic 72, no. 3 (September 2007): 755–66. http://dx.doi.org/10.2178/jsl/1191333840.

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AbstractGeneric automorphisms of certain homogeneous structures are considered, for instance, the rationals as an ordered set, the countable universal homogeneous partial order, and the random graph. Two of these cases were discussed in [7], where it was shown that there is a generic automorphism of the second in the sense introduced in [10], In this paper, I study various possible definitions of ‘generic’ and ‘mutually generic’, and discuss the existence of mutually generic automorphisms in some cases. In addition, generics in the automorphism group of the rational circular order are considered.

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26

PAPISTAS, ATHANASSIOS I. "AUTOMORPHISMS OF RELATIVELY FREE GROUPS IN THE VARIETY N2A ∧ AN2 ∧ Nc." Journal of the London Mathematical Society 63, no. 3 (June 2001): 607–22. http://dx.doi.org/10.1017/s0024610701002058.

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For positive integers n and c, with n [ges ] 2, let Gn, c be a relatively free group of finite rank n in the variety N2A ∧ AN2 ∧ Nc. It is shown that the subgroup of the automorphism group Aut(Gn, c) of Gn, c generated by the tame automorphisms and an explicitly described finite set of IA-automorphisms of Gn, c has finite index in Aut(Gn, c). Furthermore, it is proved that there are no non-trivial elements of Gn, c fixed by every tame automorphism of Gn, c.

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Amerik, Ekaterina, and Misha Verbitsky. "Construction of automorphisms of hyperkähler manifolds." Compositio Mathematica 153, no. 8 (May 31, 2017): 1610–21. http://dx.doi.org/10.1112/s0010437x17007138.

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Let $M$ be an irreducible holomorphic symplectic (hyperkähler) manifold. If $b_{2}(M)\geqslant 5$, we construct a deformation $M^{\prime }$ of $M$ which admits a symplectic automorphism of infinite order. This automorphism is hyperbolic, that is, its action on the space of real $(1,1)$-classes is hyperbolic. If $b_{2}(M)\geqslant 14$, similarly, we construct a deformation which admits a parabolic automorphism (and many other automorphisms as well).

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Urech, Christian, and Susanna Zimmermann. "Continuous automorphisms of Cremona groups." International Journal of Mathematics 32, no. 04 (February 27, 2021): 2150019. http://dx.doi.org/10.1142/s0129167x21500191.

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We show that if a group automorphism of a Cremona group of arbitrary rank is also a homeomorphism with respect to either the Zariski or the Euclidean topology, then it is inner up to a field automorphism of the base-field. Moreover, we show that a similar result holds if we consider groups of polynomial automorphisms of affine spaces instead of Cremona groups.

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Kofinas, C. E., and A. I. Papistas. "On automorphisms of free center-by-metabelian and nilpotent groups and Lie algebras." Journal of Algebra and Its Applications 17, no. 04 (April 2018): 1850077. http://dx.doi.org/10.1142/s0219498818500779.

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Let [Formula: see text] be a field of characteristic zero. For positive integers [Formula: see text] and [Formula: see text], with [Formula: see text], let [Formula: see text] be a free center-by-metabelian and nilpotent Lie algebra over [Formula: see text] of rank [Formula: see text] and class [Formula: see text], freely generated by a set [Formula: see text]. It is shown that the automorphism group [Formula: see text] of [Formula: see text] is generated by the general linear group [Formula: see text] and two more IA-automorphisms. Let [Formula: see text] be the field of rational numbers. We give [Formula: see text] the structure of a group, say [Formula: see text], via the Baker–Campbell–Hausdorff formula. Let [Formula: see text] be the subgroup of [Formula: see text] generated by [Formula: see text]. We prove that the subgroup of [Formula: see text] generated by the tame automorphisms [Formula: see text] and three more IA-automorphisms of [Formula: see text] has finite index in [Formula: see text]. For [Formula: see text], the subgroup of [Formula: see text] generated by the tame automorphisms [Formula: see text] and two more IA-automorphisms of [Formula: see text] has finite index in [Formula: see text]. A similar result is proved for the automorphism group of a free center-by-metabelian and nilpotent group of rank [Formula: see text] and class [Formula: see text].

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30

Blossier, Thomas. "Automorphism groups of trivial strongly minimal structures." Journal of Symbolic Logic 68, no. 2 (June 2003): 644–68. http://dx.doi.org/10.2178/jsl/1052669069.

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AbstractWe study automorphism groups of trivial strongly minimal structures. First we give a characterization of structures of bounded valency through their groups of automorphisms. Then we characterize the triplets of groups which can be realized as the automorphism group of a non algebraic component, the subgroup stabilizer of a point and the subgroup of strong automorphisms in a trivial strongly minimal structure, and also we give a reconstruction result. Finally, using HNN extensions we show that any profinite group can be realized as the stabilizer of a point in a strongly minimal structure of bounded valency.

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Duncan, Andrew J., and Vladimir N. Remeslennikov. "Automorphisms of partially commutative groups III: Inversions and transvections." International Journal of Algebra and Computation 28, no. 06 (September 2018): 1017–47. http://dx.doi.org/10.1142/s0218196718500455.

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The automorphism group of a partially commutative group [Formula: see text] is generated by four types of automorphism: graph automorphisms, inversions, transvections and vertex conjugating automorphisms. We find a characterisation of the subgroup [Formula: see text] generated by inversions and transvections, in terms of stabilisers of subgroups of [Formula: see text] generated by the so-called “admissible” subsets of [Formula: see text]. This is used to give a decomposition of [Formula: see text] as a chain of semi-direct products of (mostly) tractable and recognisable subgroups; which in turn gives rise to a presentation of [Formula: see text].

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Antón Sancho, Álvaro. "Automorphisms of the moduli space of principal $G$-bundles induced by outer automorphisms of $G$." MATHEMATICA SCANDINAVICA 122, no. 1 (February 20, 2018): 53. http://dx.doi.org/10.7146/math.scand.a-26348.

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In this work we study finite-order automorphisms of the moduli space of principal $G$-bundles coming from outer automorphisms of the structure group when $G$ is a simple complex Lie group. We do this by describing the subvarieties of fixed points for the action of that automorphisms on the moduli space of principal $G$-bundles. In particular, we prove that these fixed points are reductions of structure group to the subgroup of fixed points of the outer automorphism. Moreover, we study the way in which these fixed points fall into the stable or nonstable locus of the moduli.

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33

Drensky, Vesselin, and C. K. Gupta. "Automorphisms of Free Nilpotent Lie Algebras." Canadian Journal of Mathematics 42, no. 2 (April 1, 1990): 259–79. http://dx.doi.org/10.4153/cjm-1990-015-1.

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Let Fm be the free Lie algebra of rank m over a field K of characteristic 0 freely generated by the set ﹛x1,… ,xm﹜, m ≧ 2. Cohn [7] proved that the automorphism group Aut Fm of the K-algebra Fm is generated by the following automorphisms: (i) automorphisms which are induced by the action of the general linear group GLm (= GLm(K)) on the subspace of Fm spanned by ﹛x1, … ,xm﹜; (ii) automorphisms of the form x1 → x1 +f(x2,… ,xm),Xk → xk, k ≠ 1, where the polynomial f(x2,…,xm) does not depend on x1.

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Fürdös, Stefan, and Bernhard Lamel. "Regularity of infinitesimal CR automorphisms." International Journal of Mathematics 27, no. 14 (December 2016): 1650112. http://dx.doi.org/10.1142/s0129167x16501123.

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We study the regularity of infinitesimal CR automorphisms of abstract CR structures which possess a certain microlocal extension and show that there are smooth multipliers, completely determined by the CR structure, such that if [Formula: see text] is such an infinitesimal CR automorphism, then [Formula: see text] is smooth for all multipliers [Formula: see text]. As an application, we study the regularity of infinitesimal automorphisms of certain infinite type hypersurfaces in [Formula: see text].

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35

Moskowicz, Vered. "About Dixmier's conjecture." Journal of Algebra and Its Applications 14, no. 10 (September 2015): 1550140. http://dx.doi.org/10.1142/s0219498815501406.

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The well-known Dixmier conjecture [5] asks if every algebra endomorphism of the first Weyl algebra over a characteristic zero field is an automorphism. We bring a hopefully easier to solve conjecture, called the γ, δ conjecture, and show that it is equivalent to the Dixmier conjecture. In the group generated by automorphisms and anti-automorphisms of A1, all the involutions belong to one conjugacy class, hence: • Every involutive endomorphism from (A1, γ) to (A1, δ) is an automorphism (γ and δ are two involutions on A1). • Given an endomorphism f of A1 (not necessarily an involutive endomorphism), if one of f(X), f(Y) is symmetric or skew-symmetric (with respect to any involution on A1), then f is an automorphism.

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36

ATABEKYAN, V. S. "THE GROUPS OF AUTOMORPHISMS ARE COMPLETE FOR FREE BURNSIDE GROUPS OF ODD EXPONENTS n ≥ 1003." International Journal of Algebra and Computation 23, no. 06 (September 2013): 1485–96. http://dx.doi.org/10.1142/s0218196713500318.

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It is proved that the group of automorphisms Aut (B(m, n)) of the free Burnside group B(m, n) is complete for every odd exponent n ≥ 1003 and for any m > 1, that is, it has a trivial center and any automorphism of Aut (B(m, n)) is inner. Thus, the automorphism tower problem for groups B(m, n) is solved and it is showed that it is as short as the automorphism tower of the absolutely free groups. Moreover, the group of all inner automorphisms Inn (B(m, n)) is the unique normal subgroup in Aut (B(m, n)) among all its subgroups, which are isomorphic to free Burnside group B(s, n) of some rank s.

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37

Krstić, Sava, Martin Lustig, and Karen Vogtmann. "AN EQUIVARIANT WHITEHEAD ALGORITHM AND CONJUGACY FOR ROOTS OF DEHN TWIST AUTOMORPHISMS." Proceedings of the Edinburgh Mathematical Society 44, no. 1 (February 2001): 117–41. http://dx.doi.org/10.1017/s0013091599000061.

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AbstractGiven finite sets of cyclic words $\{u_1,\dots,u_k\}$ and $\{v_1,\dots,v_k\}$ in a finitely generated free group $F$ and two finite groups $A$ and $B$ of outer automorphisms of $F$, we produce an algorithm to decide whether there is an automorphism which conjugates $A$ to $B$ and takes $u_i$ to $v_i$ for each $i$. If $A$ and $B$ are trivial, this is the classic algorithm due to Whitehead. We use this algorithm together with Cohen and Lustig’s solution to the conjugacy problem for Dehn twist automorphisms of $F$ to solve the conjugacy problem for outer automorphisms which have a power which is a Dehn twist. This settles the conjugacy problem for all automorphisms of $F$ which have linear growth.AMS 2000 Mathematics subject classification: Primary 20F32. Secondary 57M07

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FEWSTER, CHRISTOPHER J. "ENDOMORPHISMS AND AUTOMORPHISMS OF LOCALLY COVARIANT QUANTUM FIELD THEORIES." Reviews in Mathematical Physics 25, no. 05 (June 2013): 1350008. http://dx.doi.org/10.1142/s0129055x13500086.

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In the framework of locally covariant quantum field theory, a theory is described as a functor from a category of spacetimes to a category of *-algebras. It is proposed that the global gauge group of such a theory can be identified as the group of automorphisms of the defining functor. Consequently, multiplets of fields may be identified at the functorial level. It is shown that locally covariant theories that obey standard assumptions in Minkowski space, including energy compactness, have no proper endomorphisms (i.e. all endomorphisms are automorphisms) and have a compact automorphism group. Further, it is shown how the endomorphisms and automorphisms of a locally covariant theory may, in principle, be classified in any single spacetime. As an example, the endomorphisms and automorphisms of a system of finitely many free scalar fields are completely classified.

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39

Shirvani, M. "The finite inner automorphism groups of division rings." Mathematical Proceedings of the Cambridge Philosophical Society 118, no. 2 (September 1995): 207–13. http://dx.doi.org/10.1017/s030500410007359x.

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Let G be a finite group of automorphisms of an associative ring R. Then the inner automorphisms (x↦ u−1xu = xu, for some unit u of R) contained in G form a normal subgroup G0 of G. In general, the Galois theory associated with the outer automorphism group G/G0 is quit well behaved (e.g. [7], 2·3–2·7, 2·10), while little group-theoretic restriction on the structure of G/G0 may be expected (even when R is a commutative field). The structure of the inner automorphism groups G0 does not seem to have received much attention so far. Here we classify the finite groups of inner automorphisms of division rings, i.e. the finite subgroups of PGL (1, D), where D is a division ring. Such groups also arise in the study of finite collineation groups of projective spaces (via the fundamental theorem of projective geometry, cf. [1], 2·26), and provide examples of finite groups having faithful irreducible projective representations over fields.

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Öztekin, Özge, and Naime Ekici. "Central automorphisms of free nilpotent Lie algebras." Journal of Algebra and Its Applications 16, no. 11 (October 4, 2017): 1750205. http://dx.doi.org/10.1142/s021949881750205x.

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Let [Formula: see text] be the free nilpotent Lie algebra of finite rank [Formula: see text] [Formula: see text] and nilpotency class [Formula: see text] over a field of characteristic zero. We give a characterization of central automorphisms of [Formula: see text] and we find sufficient conditions for an automorphism of [Formula: see text] to be a central automorphism.

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41

Collins, D. J., and E. C. Turner. "All automorphisms of free groups with maximal rank fixed subgroups." Mathematical Proceedings of the Cambridge Philosophical Society 119, no. 4 (May 1996): 615–30. http://dx.doi.org/10.1017/s0305004100074466.

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The Scott Conjecture, proven by Bestvina and Handel [2] says that an automorphism of a free group of rank n has fixed subgroup of rank at most n. We characterise in Theorem A below those automorphisms that realise this maximum. It follows from this characterisation, for example, that any such automorphism has linear growth. In our paper [3], we generalised the Scott Conjecture to arbitrary free products, using Kuros rank (see Section 2 below) in place of free rank; in Theorem B, we characterise those automorphisms of a free product realising the maximum. We show that in this case the growth rate is also linear. These results extend those of [4].

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Bryant, R. M., and C. K. Gupta. "Automorphisms of free nilpotent-by-abelian groups." Mathematical Proceedings of the Cambridge Philosophical Society 114, no. 1 (July 1993): 143–47. http://dx.doi.org/10.1017/s0305004100071474.

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Let Fn be a free group of finite rank n with basis {x1,…, xn}. Let be a variety of groups and write for the verbal subgroup of Fn corresponding to . (See [11] for information on varieties and related concepts.) Every automorphism of Fn induces an automorphism of the relatively free group Fn/V, and those automorphisms of Fn/V arising in this way are called tame. If is the variety of all metabelian groups and n ╪ 3 then every automorphism of Fn/V is tame [2, 4, 12]. But this is an exceptional situation. For many (and probably most) other varieties , Fn/V has non-tame automorphisms for all sufficiently large n. This holds for the variety of all nilpotent groups of class at most c where c ≥ 3 [1, 3] and for nearly all product varieties including, in particular, the variety of all groups whose derived groups are nilpotent of class at most c, where c > 2 [10, 13].

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43

Driss, Aiat Hadj Ahmed, and Ben Yakoub l'Moufadal. "Jordan automorphisms, Jordan derivations of generalized triangular matrix algebra." International Journal of Mathematics and Mathematical Sciences 2005, no. 13 (2005): 2125–32. http://dx.doi.org/10.1155/ijmms.2005.2125.

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We investigate Jordan automorphisms and Jordan derivations of a class of algebras called generalized triangular matrix algebras. We prove that any Jordan automorphism on such an algebra is either an automorphism or an antiautomorphism and any Jordan derivation on such an algebra is a derivation.

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44

Wang, Li. "Automorphism Groups of the Imprimitive Complex Reflection Groups II." Algebra Colloquium 18, no. 02 (June 2011): 315–26. http://dx.doi.org/10.1142/s1005386711000216.

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We prove that the automorphism group Aut (m,p,n) of an imprimitive complex reflection group G(m,p,n) is the product of a normal subgroup T(m,p,n) by a subgroup R(m,p,n), where R(m,p,n) is the group of automorphisms that preserve reflections and T(m,p,n) consists of automorphisms that map every element of G(m,p,n) to a scalar multiple of itself.

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PAPISTAS, ATHANASSIOS I. "ON AUTOMORPHISMS OF FREE NILPOTENT-BY-ABELIAN GROUPS." International Journal of Algebra and Computation 16, no. 05 (October 2006): 827–37. http://dx.doi.org/10.1142/s021819670600330x.

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For a positive integer n, with n ≥ 4, let Gn be a free nilpotent of class 2-by-abelian group of rank n. In this paper, it is shown that the automorphism group of Gn is generated by the tame automorphisms of Gn and an explicitly given set of non-tame IA-automorphisms of Gn. This extends a result of Gupta and Levin [9]. Furthermore, the aforementioned result extends a result of Stöhr [13].

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46

Bavard, Juliette, Spencer Dowdall, and Kasra Rafi. "Isomorphisms Between Big Mapping Class Groups." International Mathematics Research Notices 2020, no. 10 (May 25, 2018): 3084–99. http://dx.doi.org/10.1093/imrn/rny093.

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Abstract We show that any isomorphism between mapping class groups of orientable infinite-type surfaces is induced by a homeomorphism between the surfaces. Our argument additionally applies to automorphisms between finite-index subgroups of these “big” mapping class groups and shows that each finite-index subgroup has finite outer automorphism group. As a key ingredient, we prove that all simplicial automorphisms between curve complexes of infinite-type orientable surfaces are induced by homeomorphisms.

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SHI, JIAN-YI, and LI WANG. "AUTOMORPHISM GROUPS OF THE IMPRIMITIVE COMPLEX REFLECTION GROUPS." Journal of the Australian Mathematical Society 86, no. 1 (February 2009): 123–38. http://dx.doi.org/10.1017/s1446788708000748.

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48

DOSE, VALERIO. "ON THE AUTOMORPHISMS OF THE NONSPLIT CARTAN MODULAR CURVES OF PRIME LEVEL." Nagoya Mathematical Journal 224, no. 1 (September 9, 2016): 74–92. http://dx.doi.org/10.1017/nmj.2016.32.

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We study the automorphisms of the nonsplit Cartan modular curves $X_{\text{ns}}(p)$ of prime level $p$. We prove that if $p\geqslant 29$ all the automorphisms preserve the cusps. Furthermore, if $p\equiv 1~\text{mod}~12$ and $p\neq 13$, the automorphism group is generated by the modular involution given by the normalizer of a nonsplit Cartan subgroup of $\text{GL}_{2}(\mathbb{F}_{p})$. We also prove that for every $p\geqslant 29$ the existence of an exceptional rational automorphism would give rise to an exceptional rational point on the modular curve $X_{\text{ns}}^{+}(p)$ associated to the normalizer of a nonsplit Cartan subgroup of $\text{GL}_{2}(\mathbb{F}_{p})$.

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49

YAPTI ÖZKURT, Zeynep. "Bases of fixed point subalgebras on nilpotent Leibniz algebras." Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi 26, no. 1 (January 6, 2024): 272–78. http://dx.doi.org/10.25092/baunfbed.1332488.

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Let K be a field of characteristic zero, X={x_(1,) x_2,…,x_n} and R_m={r_(1,) ,…,r_m} be two sets of variables, F be the free left nitpotent Leibniz algebra generated by X, and K[R_m ] be the commutative polynomial algebra generated by R_m over the base field K. The fixed point subalgebra of an automorphism φ is the subalgebra of F consisting of elements that are invariant under the automorphism. In this work, we consider specific automorphisms of F and determine the fixed point subalgebras of these automorphisms. Then, we find bases of these fixed point subalgebras. In addition, we get generators of these subalgebras as a free K[R_m ] -module.

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NAKAMURA, HIDEKI. "Aperiodic automorphisms of nuclear purely infinite simple $C^{\ast}$-algebras." Ergodic Theory and Dynamical Systems 20, no. 6 (December 2000): 1749–65. http://dx.doi.org/10.1017/s0143385700000973.

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We show that if two aperiodic automorphisms of a separable nuclear unital purely infinite simple $C^{\ast}$-algebra are asymptotically unitarily equivalent, then they are outer conjugate with respect to an automorphism which is isotopic to the identity automorphism. Thus, by Kirchberg and Phillips, they have the same KK-class if and only if they are outer conjugate with respect to an automorphism which is in the KK-class of the identity automorphism.

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