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Search: All articles in the CMB digital archive with keyword automorphism group

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1. CMB Online first

Schmidt, Simon; Weber, Moritz
Quantum symmetries of graph $C^*$-algebras
The study of graph $C^*$-algebras has a long history in operator algebras. Surprisingly, their quantum symmetries have never been computed so far. We close this gap by proving that the quantum automorphism group of a finite, directed graph without multiple edges acts maximally on the corresponding graph $C^*$-algebra. This shows that the quantum symmetry of a graph coincides with the quantum symmetry of the graph $C^*$-algebra. In our result, we use the definition of quantum automorphism groups of graphs as given by Banica in 2005. Note that Bichon gave a different definition in 2003; our action is inspired from his work. We review and compare these two definitions and we give a complete table of quantum automorphism groups (with respect to either of the two definitions) for undirected graphs on four vertices.

Keywords:finite graph, graph automorphism, automorphism group, quantum automorphism, graph C*-algebra, quantum group, quantum symmetry
Categories:46LXX, 05CXX, 20B25

2. CMB 2016 (vol 59 pp. 346)

Krantz, Steven
On a Theorem of Bers, with Applications to the Study of Automorphism Groups of Domains
We study and generalize a classical theorem of L. Bers that classifies domains up to biholomorphic equivalence in terms of the algebras of holomorphic functions on those domains. Then we develop applications of these results to the study of domains with noncompact automorphism group.

Keywords:Bers's theorem, algebras of holomorphic functions, noncompact automorphism group, biholomorphic equivalence
Categories:32A38, 30H50, 32A10, 32M99

3. CMB 2014 (vol 58 pp. 196)

Yang, Qingjie; Zhong, Weiting
Dihedral Groups of order $2p$ of Automorphisms of Compact Riemann Surfaces of Genus $p-1$
In this paper we prove that there is only one conjugacy class of dihedral group of order $2p$ in the $2(p-1)\times 2(p-1)$ integral symplectic group can be realized by an analytic automorphism group of compact connected Riemann surfaces of genus $p-1$. A pair of representative generators of the realizable class is also given.

Keywords:dihedral group, automorphism group, Riemann surface, integral symplectic matrix, fundamental domain
Categories:20H25, 57M60

4. CMB 2009 (vol 52 pp. 366)

Gévay, Gábor
A Class of Cellulated Spheres with Non-Polytopal Symmetries
We construct, for all $d\geq 4$, a cellulation of $\mathbb S^{d-1}$. We prove that these cellulations cannot be polytopal with maximal combinatorial symmetry. Such non-realizability phenomenon was first described in dimension 4 by Bokowski, Ewald and Kleinschmidt, and, to the knowledge of the author, until now there have not been any known examples in higher dimensions. As a starting point for the construction, we introduce a new class of (Wythoffian) uniform polytopes, which we call duplexes. In proving our main result, we use some tools that we developed earlier while studying perfect polytopes. In particular, we prove perfectness of the duplexes; furthermore, we prove and make use of the perfectness of another new class of polytopes which we obtain by a variant of the so-called $E$-construction introduced by Eppstein, Kuperberg and Ziegler.

Keywords:CW sphere, polytopality, automorphism group, symmetry group, uniform polytope
Categories:52B11, 52B15, 52B70

5. CMB 2007 (vol 50 pp. 206)

Golasiński, Marek; Gonçalves, Daciberg Lima
Spherical Space Forms: Homotopy Types and Self-Equivalences for the Group $({\mathbb Z}/a\rtimes{\mathbb Z}/b) \times SL_2\,(\mathbb{F}_p)$
Let $G=({\mathbb Z}/a\rtimes{\mathbb Z}/b) \times \SL_2(\mathbb{F}_p)$, and let $X(n)$ be an $n$-dimensional $CW$-complex of the homotopy type of an $n$-sphere. We study the automorphism group $\Aut (G)$ in order to compute the number of distinct homotopy types of spherical space forms with respect to free and cellular $G$-actions on all $CW$-complexes $X(2dn-1)$, where $2d$ is the period of $G$. The groups ${\mathcal E}(X(2dn-1)/\mu)$ of self homotopy equivalences of space forms $X(2dn-1)/\mu$ associated with free and cellular $G$-actions $\mu$ on $X(2dn-1)$ are determined as well.

Keywords:automorphism group, $CW$-complex, free and cellular $G$-action, group of self homotopy equivalences, Lyndon--Hochschild--Serre spectral sequence, special (linear) group, spherical space form
Categories:55M35, 55P15, 20E22, 20F28, 57S17

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