1 - Oberwolfach Preprints (OWP)
http://publications.mfo.de/handle/mfo/19
The Oberwolfach Preprints (OWP) mainly contain research results related to a longer stay in Oberwolfach. In particular, this concerns the Research in Pairs program and the Oberwolfach Leibniz Fellows, but this can also include an Oberwolfach Lecture, for example.Tue, 23 Apr 2019 05:52:04 GMT2019-04-23T05:52:04ZOn a Group Functor Describing Invariants of Algebraic Surfaces
http://publications.mfo.de/handle/mfo/1409
On a Group Functor Describing Invariants of Algebraic Surfaces
Dietrich, Heiko; Moravec, Primož
Liedtke (2008) has introduced group functors $K$ and $\tilde K$, which are used in the context of describing certain invariants for complex algebraic surfaces. He proved that these functors are connected to the theory of central extensions and Schur multipliers. In this work we relate $K$ and $\tilde K$ to a group functor $\tau$ arising in the construction of the non-abelian exterior square of a group. In contrast to $\tilde K$, there exist efficient algorithms for constructing $\tau$, especially for polycyclic groups. Supported by computations with the computer algebra system GAP, we investigate when $K(G,3)$ is a quotient of $\tau(G)$, and when $\tau(G)$ and $\tilde K(G,3)$ are isomorphic.
Fri, 01 Mar 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/14092019-03-01T00:00:00ZDietrich, HeikoMoravec, PrimožLiedtke (2008) has introduced group functors $K$ and $\tilde K$, which are used in the context of describing certain invariants for complex algebraic surfaces. He proved that these functors are connected to the theory of central extensions and Schur multipliers. In this work we relate $K$ and $\tilde K$ to a group functor $\tau$ arising in the construction of the non-abelian exterior square of a group. In contrast to $\tilde K$, there exist efficient algorithms for constructing $\tau$, especially for polycyclic groups. Supported by computations with the computer algebra system GAP, we investigate when $K(G,3)$ is a quotient of $\tau(G)$, and when $\tau(G)$ and $\tilde K(G,3)$ are isomorphic.Weighted Surface Algebras: General Version
http://publications.mfo.de/handle/mfo/1408
Weighted Surface Algebras: General Version
Erdmann, Karin; Skowroński, Andrzej
We introduce general weighted surface algebras of triangulated surfaces with arbitrarily oriented triangles and describe their basic properties. In particular, we prove that all these algebras, except the singular disc, triangle, tetrahedral and spherical algebras, are symmetric tame periodic algebras of period 4.
Thu, 28 Feb 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/14082019-02-28T00:00:00ZErdmann, KarinSkowroński, AndrzejWe introduce general weighted surface algebras of triangulated surfaces with arbitrarily oriented triangles and describe their basic properties. In particular, we prove that all these algebras, except the singular disc, triangle, tetrahedral and spherical algebras, are symmetric tame periodic algebras of period 4.Group Algebras of Compact Groups. A New Way of Producing Group Hopf Algebras over Real and Complex Fields: Weakly Complete Topological Vector Spaces
http://publications.mfo.de/handle/mfo/1407
Group Algebras of Compact Groups. A New Way of Producing Group Hopf Algebras over Real and Complex Fields: Weakly Complete Topological Vector Spaces
Hofmann, Karl Heinrich; Kramer, Linus
Weakly complete real or complex associative algebras $A$ are necessarily projective limits of finite dimensional algebras. Their group of units $A^{-1}$ is a pro-Lie group with the associated topological Lie algebra $A_{\rm Lie}$ of $A$ as Lie algebra and the globally defined exponential function $\exp\colon A\to A^{-1}$ as the exponential function of $A^{-1}$. With each topological group $G$, a weakly complete group algebra $\mathbb K[G]$ is associated functorially so that the functor $G\mapsto \mathbb K[G]$ is left adjoint to $A\mapsto A^{-1}$. The group algebra $\mathbb K[G]$ is a weakly complete Hopf algebra. If $G$ is compact, then $\mathbb R[G]$ contains $G$ as the set of grouplike elements.
The category of all real weakly complete Hopf algebras $A$ with a compact group of grouplike elements whose linear span is dense in $A$ is equivalent to the category of compact groups. The group algebra $A=\mathbb R[G]$ of a compact group $G$ contains a copy of the Lie algebra $\mathfrak L(G)$ in $A_{\rm Lie}$; it also contains all probability measures on $G$. The dual of the group algebra $\mathbb R[G]$ is the Hopf algebra ${\cal R}(G,\mathbb R)$ of representative functions of $G$. The rather straightforward duality between vector spaces and weakly complete vector spaces thus becomes the basis
of a duality ${\cal R}(G,\mathbb R)\leftrightarrow \mathbb R[G]$ and thus yields a new aspect of Tannaka duality. In the case of a compact abelian $G$, an alternative concrete construction of $\mathbb K[G]$ is given both for $\mathbb K=\mathbb C$ and $\mathbb K=\mathbb R$. Because of the presence of $\mathfrak L(G)$, the enveloping algebra of weakly complete Lie algebras are introduced and placed into relation with $\mathbb K[G]$.
Wed, 27 Feb 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/14072019-02-27T00:00:00ZHofmann, Karl HeinrichKramer, LinusWeakly complete real or complex associative algebras $A$ are necessarily projective limits of finite dimensional algebras. Their group of units $A^{-1}$ is a pro-Lie group with the associated topological Lie algebra $A_{\rm Lie}$ of $A$ as Lie algebra and the globally defined exponential function $\exp\colon A\to A^{-1}$ as the exponential function of $A^{-1}$. With each topological group $G$, a weakly complete group algebra $\mathbb K[G]$ is associated functorially so that the functor $G\mapsto \mathbb K[G]$ is left adjoint to $A\mapsto A^{-1}$. The group algebra $\mathbb K[G]$ is a weakly complete Hopf algebra. If $G$ is compact, then $\mathbb R[G]$ contains $G$ as the set of grouplike elements.
The category of all real weakly complete Hopf algebras $A$ with a compact group of grouplike elements whose linear span is dense in $A$ is equivalent to the category of compact groups. The group algebra $A=\mathbb R[G]$ of a compact group $G$ contains a copy of the Lie algebra $\mathfrak L(G)$ in $A_{\rm Lie}$; it also contains all probability measures on $G$. The dual of the group algebra $\mathbb R[G]$ is the Hopf algebra ${\cal R}(G,\mathbb R)$ of representative functions of $G$. The rather straightforward duality between vector spaces and weakly complete vector spaces thus becomes the basis
of a duality ${\cal R}(G,\mathbb R)\leftrightarrow \mathbb R[G]$ and thus yields a new aspect of Tannaka duality. In the case of a compact abelian $G$, an alternative concrete construction of $\mathbb K[G]$ is given both for $\mathbb K=\mathbb C$ and $\mathbb K=\mathbb R$. Because of the presence of $\mathfrak L(G)$, the enveloping algebra of weakly complete Lie algebras are introduced and placed into relation with $\mathbb K[G]$.Hölder Continuity of the Spectra for Aperiodic Hamiltonians
http://publications.mfo.de/handle/mfo/1406
Hölder Continuity of the Spectra for Aperiodic Hamiltonians
Beckus, Siegfried; Bellissard, Jean; Cornean, Horia
We study the spectral location of a strongly pattern equivariant Hamiltonians arising through configurations on a colored lattice. Roughly speaking, two configurations are "close to each other" if, up to a translation, they "almost coincide" on a large fixed ball. The larger this ball is, the more similar they are, and this induces a metric on the space of the corresponding dynamical systems. Our main result states that the map which sends a given configuration into the spectrum of its associated Hamiltonian, is Hölder (even Lipschitz) continuous in the usual Hausdorff metric. Specifically, the spectral distance of two Hamiltonians is estimated by the distance of the corresponding dynamical systems.
Tue, 26 Feb 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/14062019-02-26T00:00:00ZBeckus, SiegfriedBellissard, JeanCornean, HoriaWe study the spectral location of a strongly pattern equivariant Hamiltonians arising through configurations on a colored lattice. Roughly speaking, two configurations are "close to each other" if, up to a translation, they "almost coincide" on a large fixed ball. The larger this ball is, the more similar they are, and this induces a metric on the space of the corresponding dynamical systems. Our main result states that the map which sends a given configuration into the spectrum of its associated Hamiltonian, is Hölder (even Lipschitz) continuous in the usual Hausdorff metric. Specifically, the spectral distance of two Hamiltonians is estimated by the distance of the corresponding dynamical systems.Applications of BV Type Spaces
http://publications.mfo.de/handle/mfo/1403
Applications of BV Type Spaces
Appell, Jürgen; Bugajewska, Daria; Kasprzak, Piotr; Merentes, Nelson; Reinwand, Simon; Sánchez, José Luis
Wed, 13 Feb 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/14032019-02-13T00:00:00ZAppell, JürgenBugajewska, DariaKasprzak, PiotrMerentes, NelsonReinwand, SimonSánchez, José LuisTime Discretization Schemes for Hyperbolic Systems on Networks by ε-Expansion
http://publications.mfo.de/handle/mfo/1402
Time Discretization Schemes for Hyperbolic Systems on Networks by ε-Expansion
Altmann, Robert; Zimmer, Christoph
We consider partial differential equations on networks with a small parameter $\epsilon$, which are hyperbolic for $\epsilon>0$ and parabolic for $\epsilon=0$. With a combination of an $\epsilon$-expansion and Runge-Kutta schemes for constrained systems of parabolic type, we derive a new class of time discretization schemes for hyperbolic systems on networks, which are constrained due to interconnection conditions. For the analysis we consider the coupled system equations as partial differential-algebraic equations based on the variational formulation of the problem. We discuss well-posedness of the resulting systems and estimate the error caused by the $\epsilon$-expansion.
Tue, 12 Feb 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/14022019-02-12T00:00:00ZAltmann, RobertZimmer, ChristophWe consider partial differential equations on networks with a small parameter $\epsilon$, which are hyperbolic for $\epsilon>0$ and parabolic for $\epsilon=0$. With a combination of an $\epsilon$-expansion and Runge-Kutta schemes for constrained systems of parabolic type, we derive a new class of time discretization schemes for hyperbolic systems on networks, which are constrained due to interconnection conditions. For the analysis we consider the coupled system equations as partial differential-algebraic equations based on the variational formulation of the problem. We discuss well-posedness of the resulting systems and estimate the error caused by the $\epsilon$-expansion.A Function Algebra Providing New Mergelyan Type Theorems in Several Complex Variables
http://publications.mfo.de/handle/mfo/1401
A Function Algebra Providing New Mergelyan Type Theorems in Several Complex Variables
Falcó, Javier; Gauthier, Paul Montpetit; Manolaki, Myrto; Nestoridis, Vassili
For compact sets $K\subset \mathbb C^{d}$, we introduce a subalgebra $A_{D}(K)$ of $A(K)$, which allows us to obtain Mergelyan type theorems for products of planar compact sets as well as for graphs of functions.
Mon, 11 Feb 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/14012019-02-11T00:00:00ZFalcó, JavierGauthier, Paul MontpetitManolaki, MyrtoNestoridis, VassiliFor compact sets $K\subset \mathbb C^{d}$, we introduce a subalgebra $A_{D}(K)$ of $A(K)$, which allows us to obtain Mergelyan type theorems for products of planar compact sets as well as for graphs of functions.Cataland: Why the Fuß?
http://publications.mfo.de/handle/mfo/1398
Cataland: Why the Fuß?
Stump, Christian; Thomas, Hugh; Williams, Nathan
The three main objects in noncrossing Catalan combinatorics associated to a finite Coxeter system are noncrossing partitions, clusters, and sortable elements. The first two of these have known Fuß-Catalan generalizations. We provide new viewpoints for both and introduce the missing generalization of sortable elements by lifting the theory from the Coxeter system to the associated positive Artin monoid. We show how this new perspective ties together all three generalizations, providing a uniform framework for noncrossing Fuß-Catalan combinatorics. Having developed the combinatorial theory, we provide an interpretation of our generalizations in the language of the representation theory of hereditary Artin algebras.
Mon, 21 Jan 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13982019-01-21T00:00:00ZStump, ChristianThomas, HughWilliams, NathanThe three main objects in noncrossing Catalan combinatorics associated to a finite Coxeter system are noncrossing partitions, clusters, and sortable elements. The first two of these have known Fuß-Catalan generalizations. We provide new viewpoints for both and introduce the missing generalization of sortable elements by lifting the theory from the Coxeter system to the associated positive Artin monoid. We show how this new perspective ties together all three generalizations, providing a uniform framework for noncrossing Fuß-Catalan combinatorics. Having developed the combinatorial theory, we provide an interpretation of our generalizations in the language of the representation theory of hereditary Artin algebras.The Tutte Polynomial of Ideal Arrangements
http://publications.mfo.de/handle/mfo/1395
The Tutte Polynomial of Ideal Arrangements
Randriamaro, Hery
The Tutte polynomial is originally a bivariate polynomial enumerating the colorings of a graph and of its dual graph. But it reveals more of the internal structure of the graph like its number of forests, of spanning subgraphs, and of acyclic orientations. In 2007, Ardila extended the notion of Tutte polynomial to hyperplane arrangements, and computed the Tutte polynomials of the classical root systems for a certain prime power of the first variable. In this article, we compute the Tutte polynomials of ideal arrangements. Those arrangements were introduced in 2006 by Sommers and Tymoczko, and are defined for ideals of root systems. For the ideals of the classical root systems, we bring a slight improvement of the finite field method showing that it can applied on any finite field whose cardinality is not a minor of the matrix associated to a hyperplane arrangement. Computing the minor set associated to an ideal of a classical root system permits us particularly to deduce the Tutte polynomials of the classical root systems. For the ideals of the exceptional root systems of type G2, F4, and E6, we use the formula of Crapo.
Fri, 21 Dec 2018 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13952018-12-21T00:00:00ZRandriamaro, HeryThe Tutte polynomial is originally a bivariate polynomial enumerating the colorings of a graph and of its dual graph. But it reveals more of the internal structure of the graph like its number of forests, of spanning subgraphs, and of acyclic orientations. In 2007, Ardila extended the notion of Tutte polynomial to hyperplane arrangements, and computed the Tutte polynomials of the classical root systems for a certain prime power of the first variable. In this article, we compute the Tutte polynomials of ideal arrangements. Those arrangements were introduced in 2006 by Sommers and Tymoczko, and are defined for ideals of root systems. For the ideals of the classical root systems, we bring a slight improvement of the finite field method showing that it can applied on any finite field whose cardinality is not a minor of the matrix associated to a hyperplane arrangement. Computing the minor set associated to an ideal of a classical root system permits us particularly to deduce the Tutte polynomials of the classical root systems. For the ideals of the exceptional root systems of type G2, F4, and E6, we use the formula of Crapo.Spectral Continuity for Aperiodic Quantum Systems II. Periodic Approximations in 1D
http://publications.mfo.de/handle/mfo/1394
Spectral Continuity for Aperiodic Quantum Systems II. Periodic Approximations in 1D
Beckus, Siegfried; Bellissard, Jean; De Nittis, Giuseppe
The existence and construction of periodic approximations with convergent spectra is crucial in solid state physics for the spectral study of corresponding Schrödinger operators. In a forthcoming work [9] this task was boiled down to the existence and construction of periodic approximations of the underlying dynamical systems in the Hausdorff topology. As a result the one-dimensional systems admitting such approximations are completely classified in the present work. In addition explicit constructions are provided for dynamical systems defined by primitive substitutions covering all studied examples such as the Fibonacci sequence or the Golay-Rudin-Shapiro sequence. One main tool is the description of the Hausdorff topology by the local pattern topology on the dictionaries as well as the GAP-graphs describing the local structure. The connection of branching vertices in the GAP-graphs and defects is discussed.
Mon, 17 Dec 2018 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13942018-12-17T00:00:00ZBeckus, SiegfriedBellissard, JeanDe Nittis, GiuseppeThe existence and construction of periodic approximations with convergent spectra is crucial in solid state physics for the spectral study of corresponding Schrödinger operators. In a forthcoming work [9] this task was boiled down to the existence and construction of periodic approximations of the underlying dynamical systems in the Hausdorff topology. As a result the one-dimensional systems admitting such approximations are completely classified in the present work. In addition explicit constructions are provided for dynamical systems defined by primitive substitutions covering all studied examples such as the Fibonacci sequence or the Golay-Rudin-Shapiro sequence. One main tool is the description of the Hausdorff topology by the local pattern topology on the dictionaries as well as the GAP-graphs describing the local structure. The connection of branching vertices in the GAP-graphs and defects is discussed.