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Mikhail V. Neshchadim - One of the best experts on this subject based on the ideXlab platform.
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Automorphisms of pure braid groups
Monatshefte für Mathematik, 2018Co-Authors: Valeriy G. Bardakov, Mikhail V. Neshchadim, Mahender SinghAbstract:In this paper, we investigate the structure of the automorphism groups of pure braid groups. We prove that, for $$n>3$$ n > 3 , $${\text {Aut}}(P_n)$$ Aut ( P n ) is generated by the subgroup $${\text {Aut}}_c(P_n)$$ Aut c ( P n ) of central Automorphisms of $$P_n$$ P n , the subgroup $${\text {Aut}}(B_n)$$ Aut ( B n ) of restrictions of Automorphisms of $$B_n$$ B n on $$P_n$$ P n and one extra automorphism $$w_n$$ w n . We also investigate the lifting and extension problem for Automorphisms of some well-known exact sequences arising from braid groups, and prove that that answers are negative in most cases. Specifically, we prove that no non-trivial central automorphism of $$P_n$$ P n can be extended to an automorphism of $$B_n$$ B n .
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Automorphisms of pure braid Groups
Monatshefte für Mathematik, 2017Co-Authors: Valeriy G. Bardakov, Mikhail V. Neshchadim, Mahender SinghAbstract:In this paper, we investigate the structure of the automorphism groups of pure braid groups. We prove that, for $n>3$, $\Aut(P_n)$ is generated by the subgroup $\Aut_c(P_n)$ of central Automorphisms of $P_n$, the subgroup $\Aut(B_n)$ of restrictions of Automorphisms of $B_n$ on $P_n$ and one extra automorphism $w_n$. We also investigate the lifting and extension problem for Automorphisms of some well-known exact sequences arising from braid groups, and prove that that answers are negative in most cases. Specifically, we prove that no non-trivial central automorphism of $P_n$ can be extended to an automorphism of $B_n$.
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An example of a nonlinearizable quasicyclic subgroup in the automorphism group of the polynomial algebra
Mathematical Notes, 2015Co-Authors: Valeriy G. Bardakov, Mikhail V. NeshchadimAbstract:As is well known, every finite subgroup of the automorphism group of the polynomial algebra of rank two over a field of characteristic zero is conjugate to the subgroup of linear Automorphisms. We show that this can fail for an arbitrary periodic subgroup. We construct an example of an Abelian p-subgroup of the automorphism group of the polynomial algebra of rank two over the field of complex numbers which is not conjugate to any subgroup of linear Automorphisms.
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Example of non-linearizable quasi-cyclic subgroup of automorphism group of polynomial algebra
arXiv: Group Theory, 2015Co-Authors: Valeriy G. Bardakov, Mikhail V. NeshchadimAbstract:It is well known that every finite subgroup of automorphism group of polynomial algebra of rank 2 over the field of zero characteristic is conjugated with a subgroup of linear Automorphisms. We prove that it is not true for an arbitrary torsion subgroup. We construct an example of abelian $p$-group of automorphism of polynomial algebra of rank 2 over the field of complex numbers, which is not conjugated with a subgroup of linear Automorphisms.
Valeriy G. Bardakov - One of the best experts on this subject based on the ideXlab platform.
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Automorphisms of pure braid groups
Monatshefte für Mathematik, 2018Co-Authors: Valeriy G. Bardakov, Mikhail V. Neshchadim, Mahender SinghAbstract:In this paper, we investigate the structure of the automorphism groups of pure braid groups. We prove that, for $$n>3$$ n > 3 , $${\text {Aut}}(P_n)$$ Aut ( P n ) is generated by the subgroup $${\text {Aut}}_c(P_n)$$ Aut c ( P n ) of central Automorphisms of $$P_n$$ P n , the subgroup $${\text {Aut}}(B_n)$$ Aut ( B n ) of restrictions of Automorphisms of $$B_n$$ B n on $$P_n$$ P n and one extra automorphism $$w_n$$ w n . We also investigate the lifting and extension problem for Automorphisms of some well-known exact sequences arising from braid groups, and prove that that answers are negative in most cases. Specifically, we prove that no non-trivial central automorphism of $$P_n$$ P n can be extended to an automorphism of $$B_n$$ B n .
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Automorphisms of pure braid Groups
Monatshefte für Mathematik, 2017Co-Authors: Valeriy G. Bardakov, Mikhail V. Neshchadim, Mahender SinghAbstract:In this paper, we investigate the structure of the automorphism groups of pure braid groups. We prove that, for $n>3$, $\Aut(P_n)$ is generated by the subgroup $\Aut_c(P_n)$ of central Automorphisms of $P_n$, the subgroup $\Aut(B_n)$ of restrictions of Automorphisms of $B_n$ on $P_n$ and one extra automorphism $w_n$. We also investigate the lifting and extension problem for Automorphisms of some well-known exact sequences arising from braid groups, and prove that that answers are negative in most cases. Specifically, we prove that no non-trivial central automorphism of $P_n$ can be extended to an automorphism of $B_n$.
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An example of a nonlinearizable quasicyclic subgroup in the automorphism group of the polynomial algebra
Mathematical Notes, 2015Co-Authors: Valeriy G. Bardakov, Mikhail V. NeshchadimAbstract:As is well known, every finite subgroup of the automorphism group of the polynomial algebra of rank two over a field of characteristic zero is conjugate to the subgroup of linear Automorphisms. We show that this can fail for an arbitrary periodic subgroup. We construct an example of an Abelian p-subgroup of the automorphism group of the polynomial algebra of rank two over the field of complex numbers which is not conjugate to any subgroup of linear Automorphisms.
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Example of non-linearizable quasi-cyclic subgroup of automorphism group of polynomial algebra
arXiv: Group Theory, 2015Co-Authors: Valeriy G. Bardakov, Mikhail V. NeshchadimAbstract:It is well known that every finite subgroup of automorphism group of polynomial algebra of rank 2 over the field of zero characteristic is conjugated with a subgroup of linear Automorphisms. We prove that it is not true for an arbitrary torsion subgroup. We construct an example of abelian $p$-group of automorphism of polynomial algebra of rank 2 over the field of complex numbers, which is not conjugated with a subgroup of linear Automorphisms.
Cattaneo Alberto - One of the best experts on this subject based on the ideXlab platform.
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Automorphismes non-symplectiques des variétés symplectiques holomorphes
HAL CCSD, 2018Co-Authors: Cattaneo AlbertoAbstract:We study Automorphisms of irreducible holomorphic symplectic manifolds of type K3^[n], i.e. manifolds which are deformation equivalent to the Hilbert scheme of n points on a K3 surface, for some n > 1. In the first part of the thesis we describe the automorphism group of the Hilbert scheme of n points on a generic projective K3 surface, i.e. a K3 surface whose Picard lattice is generated by a single ample line bundle. We show that, if it is not trivial, the automorphism group is generated by a non-symplectic involution, whose existence depends on some arithmetic conditions involving the number of points n and the polarization of the surface. We also determine necessary and sufficient conditions on the Picard lattice of the Hilbert scheme for the existence of the involution.In the second part of the thesis we study non-symplectic Automorphisms of prime order on manifolds of type K3^[n]. We investigate the properties of the invariant lattice and its orthogonal complement inside the second cohomology lattice of the manifold, providing a classification of their isometry classes. We then approach the problem of constructing examples (or at least proving the existence) of manifolds of type K3^[n] with a non-symplectic automorphism inducing on cohomology each specific action in our classification. In the case of involutions, and of Automorphisms of odd prime order for n=3,4, we are able to realize all possible cases. In order to do so, we present a new non-symplectic automorphism of order three on a ten-dimensional family of Lehn-Lehn-Sorger-van Straten eightfolds of type K3^[4]. Finally, for n 1.Dans la première partie de la thèse, nous classifions les automorphismes du schéma de Hilbert de n points sur une surface K3 projective générique, dont le réseau de Picard est engendré par un fibré ample. Nous montrons que le groupe des automorphismes est soit trivial soit engendré par une involution non-symplectique et nous déterminons des conditions numériques et géométriques pour l’existence de l’involution.Dans la deuxième partie, nous étudions les automorphismes non-symplectiques d’ordre premier des variétés de type K3^[n]. Nous déterminons les propriétés du réseau invariant de l'automorphisme et de son complément orthogonal dans le deuxième réseau de cohomologie de la variété et nous classifions leurs classes d’isométrie. Dans le cas des involutions, e des automorphismes d’ordre premier impair pour n = 3, 4, nous montrons que toutes les actions en cohomologie dans notre classification sont réalisées par un automorphism non-symplectique sur une variété de type K3^[n]. Nous construisons explicitement l’immense majorité de ces automorphismes et, en particulier, nous présentons la construction d’un nouvel automorphisme d’ordre trois sur une famille de dimension dix de variétés de Lehn-Lehn-Sorger-van Straten de type K3^[4]. Pour n < 6, nous étudions aussi les espaces de modules de dimension maximal des variétés de type K3^[n] munies d’une involution non-symplectique
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Automorphismes non-symplectiques des variétés symplectiques holomorphes
2018Co-Authors: Cattaneo AlbertoAbstract:Nous allons étudier les automorphismes des variétés symplectiques holomorphes irréductibles de type K3^[n], c'est-à-dire des variétés équivalentes par déformation au schéma de Hilbert de n points sur une surface K3, pour n > 1.Dans la première partie de la thèse, nous classifions les automorphismes du schéma de Hilbert de n points sur une surface K3 projective générique, dont le réseau de Picard est engendré par un fibré ample. Nous montrons que le groupe des automorphismes est soit trivial soit engendré par une involution non-symplectique et nous déterminons des conditions numériques et géométriques pour l’existence de l’involution.Dans la deuxième partie, nous étudions les automorphismes non-symplectiques d’ordre premier des variétés de type K3^[n]. Nous déterminons les propriétés du réseau invariant de l'automorphisme et de son complément orthogonal dans le deuxième réseau de cohomologie de la variété et nous classifions leurs classes d’isométrie. Dans le cas des involutions, e des automorphismes d’ordre premier impair pour n = 3, 4, nous montrons que toutes les actions en cohomologie dans notre classification sont réalisées par un automorphism non-symplectique sur une variété de type K3^[n]. Nous construisons explicitement l’immense majorité de ces automorphismes et, en particulier, nous présentons la construction d’un nouvel automorphisme d’ordre trois sur une famille de dimension dix de variétés de Lehn-Lehn-Sorger-van Straten de type K3^[4]. Pour n 1. In the first part of the thesis we describe the automorphism group of the Hilbert scheme of n points on a generic projective K3 surface, i.e. a K3 surface whose Picard lattice is generated by a single ample line bundle. We show that, if it is not trivial, the automorphism group is generated by a non-symplectic involution, whose existence depends on some arithmetic conditions involving the number of points n and the polarization of the surface. We also determine necessary and sufficient conditions on the Picard lattice of the Hilbert scheme for the existence of the involution.In the second part of the thesis we study non-symplectic Automorphisms of prime order on manifolds of type K3^[n]. We investigate the properties of the invariant lattice and its orthogonal complement inside the second cohomology lattice of the manifold, providing a classification of their isometry classes. We then approach the problem of constructing examples (or at least proving the existence) of manifolds of type K3^[n] with a non-symplectic automorphism inducing on cohomology each specific action in our classification. In the case of involutions, and of Automorphisms of odd prime order for n=3,4, we are able to realize all possible cases. In order to do so, we present a new non-symplectic automorphism of order three on a ten-dimensional family of Lehn-Lehn-Sorger-van Straten eightfolds of type K3^[4]. Finally, for n < 6 we describe deformation families of large dimension of manifolds of type K3^[n] equipped with a non-symplectic involution
Ralf Schiffler - One of the best experts on this subject based on the ideXlab platform.
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Cluster Automorphisms and quasi-Automorphisms
Advances in Applied Mathematics, 2019Co-Authors: Wen Chang, Ralf SchifflerAbstract:Abstract We study the relation between the cluster Automorphisms and the quasi-Automorphisms of a cluster algebra A . We prove that under some mild condition, satisfied for example by every skew-symmetric cluster algebra, the quasi-automorphism group of A is isomorphic to a subgroup of the cluster automorphism group of A t r i v , and the two groups are isomorphic if A has principal or universal coefficients; here A t r i v is the cluster algebra with trivial coefficients obtained from A by setting all frozen variables equal to the integer 1. We also compute the quasi-automorphism group of all finite type and all skew-symmetric affine type cluster algebras, and show in which types it is isomorphic to the cluster automorphism group of A t r i v .
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Cluster Automorphisms and quasi-Automorphisms
arXiv: Rings and Algebras, 2018Co-Authors: Wen Chang, Ralf SchifflerAbstract:We study the relation between the cluster Automorphisms and the quasi-Automorphisms of a cluster algebra $\mathcal{A}$. We proof that under some mild condition, satisfied for example by every skew-symmetric cluster algebra, the quasi-automorphism group of $\mathcal{A}$ is isomorphic to a subgroup of the cluster automorphism group of $\mathcal{A}_{triv}$, and the two groups are isomorphic if $\mathcal{A}$ has principal or universal coefficients; here $\mathcal{A}_{triv}$ is the cluster algebra with trivial coefficients obtained from $\mathcal{A}$ by setting all frozen variables equal to the integer 1. We also compute the quasi-automorphism group of all finite type and all skew-symmetric affine type cluster algebras, and show in which types it is isomorphic to the cluster automorphism group of $\mathcal{A}_{triv}$.
Kyounghee Kim - One of the best experts on this subject based on the ideXlab platform.
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continuous families of rational surface Automorphisms with positive entropy
Mathematische Annalen, 2010Co-Authors: Eric Bedford, Kyounghee KimAbstract:For any k we construct k-parameter families of rational surface Automorphisms with positive entropy. These are Automorphisms of surfaces \({\mathcal{X}}\), which are constructed from iterated blowups over the projective plane. In certain cases we are able to determine the exact automorphism group of \({\mathcal{X}}\), as well as when two of these surfaces are inequivalent.
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continuous families of rational surface Automorphisms with positive entropy
arXiv: Complex Variables, 2008Co-Authors: Eric Bedford, Kyounghee KimAbstract: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 exact automorphism group of X, as well as when two of the surfaces X are inequivalent.