Some notes on Pontryagin duality of abelian topological groups
We consider several questions related to Pontryagin duality in the category of abelian pro-Lie groups.
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arXiv subjects
Linus Kramer
Publications and source records attributed to Linus Kramer.
Fibrations and coset spaces for locally compact groups
Let $G$ be a topological group and let $K,L\subseteq G$ be closed subgroups, with $K\subseteq L$. We prove that if $L$ is a locally compact pro-Lie group, then the map $q:G/K\to G/L$ is a fibration. As an application of this, we obtain two older results by Skljarenko, Madison and Mostert.
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Hodge operators and groups of isometries of diagonalizable symmetric bilinear forms in characteristic two
We study groups of isometries on non-alternating symmetric bilinear forms on vector spaces of characteristic two, and actions of these groups on exterior powers of the space, viewed as modules over algebras generated by Hodge operators.
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Hodge Operators and Exceptional Isomorphisms between Unitary Groups
We give a generalization of the Hodge operator to spaces $(V,h)$ endowed with a hermitian or symmetric bilinear form $h$ over arbitrary fields, including the characteristic two case. Suitable exterior powers of $V$ become free modules over an algebra $K$ defined using such an operator. This leads to several exceptional homomorphisms from unitary groups (with respect to $h$) into groups of semi-similitudes with respect to a suitable form over some subfield of $K$. The algebra $K$ depends on $h$; it is a composition algebra unless $h$ is symmetric and the characteristic is two.
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A note on commutators in compact semisimple Lie algebras
Given two elements $A,B$ in a compact semisimple Lie algebra, we show that there is a regular element $X$ and elements $Y,Z$ with $A=[X,Y]$ and $B=[X,Z]$. In the course of the proof we show also that every linear subspace $V$ of codimension at most 2 in the Lie algebra contains a CSA.
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Weyl Tensors, Strongly Regular Graphs, Multiplicative Characters, and a Quadratic Matrix Equation
We study solutions of a quadratic matrix equation arising in Riemannian geometry. Let $S$ be a real symmetric $n\times n$-matrix with zeros on the diagonal and let $\theta$ be a real number. We construct nonzero solutions $(S,\theta)$ of the set of quadratic equations \[\sum_kS_{i,k}=0\quad\text{ and }\quad\sum_{k}S_{i,k}S_{k,j}+S_{i,j}^2=\theta S_{i,j}\text { for }i<j.\] Our solutions relate the equations to strongly regular graphs, to group rings, and to multiplicative characters of finite fields.
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Some remarks on proper actions, proper metric spaces, and buildings
We discuss various aspects of isometric group actions on proper metric spaces. As one application, we show that a proper and Weyl transitive action on a euclidean building is strongly transitive on the maximal atlas (the complete apartment system) of the building.
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On Weakly Complete Universal Enveloping Algebras: A Poincare-Birkhoff-Witt Theorem
We study universal enveloping Hopf algebras of Lie algebras in the category of weakly complete vector spaces over the real and complex field.
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Polyhedral compactifications, I
In this work we describe horofunction compactifications of metric spaces and finite dimensional real vector spaces through asymmetric metrics and asymmetric polyhedral norms by means of nonstandard methods, that is, ultrapowers of the spaces at hand. The polyhedral compactifications of the vector spaces carry the structure of stratified spaces with the strata indexed by dual faces of the polyhedral unit ball. Explicit neighborhood bases and descriptions of the horofunctions are provided.
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On weakly complete group algebras of compact groups
We study group algebras for compact groups in the category of real and complex weakly complete vector spaces. We also show that the group algebra is a quotient of the weakly complete universal enveloping algebra of the Lie algebra of the compact group. We relate this to Tannaka duality, and to functorial properties of the group algebra. We determine the structure of the group algebra in terms of the irreducible representation, both in the real and the complex case. The particular case of a compact abelian group is worked out in detail.
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Abstract homomorphisms from locally compact groups to discrete groups
We show that every abstract homomorphism $\varphi$ from a locally compact group $L$ to a graph product $G_\Gamma$, endowed with the discrete topology, is either continuous or $\varphi(L)$ lies in a 'small' parabolic subgroup. In particular, every locally compact group topology on a graph product whose graph is not 'small' is discrete. This extends earlier work by Morris-Nickolas. We also show the following. If $L$ is a locally compact group and if $G$ is a discrete group which contains no infinite torsion group and no infinitely generated abelian group, then every abstract homomorphism $\varphi:L\to G$ is either continuous, or $\varphi(L)$ is contained in the normalizer of a finite nontrivial subgroup of $G$. As an application we obtain results concerning the continuity of homomorphisms from locally compact groups to Artin and Coxeter groups.
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A new product on 2x2 matrices
We study a bilinear multiplication rule on 2x2 matrices which is intermediate between the ordinary matrix product and the Hadamard matrix product, and we relate this to the hyperbolic motion group of the plane.
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Automatic continuity of abstract homomorphisms between locally compact and Polish groups
We are concerned with questions of the following type. Suppose that $G$ and $K$ are topological groups belonging to a certain class $\cal K$ of spaces, and suppose that $\phi:K \to G$ is an abstract (i.e. not necessarily continuous) surjective group homomorphism. Under what conditions on the group $G$ and the kernel is the homomorphism $\phi$ automatically continuous and open? Questions of this type have a long history and were studied in particular for the case that $G$ and $K$ are Lie groups, compact groups, or Polish groups. We develop an axiomatic approach, which allows us to resolve the question uniformly for different classes of topological groups. In this way we are able to extend the classical results about automatic continuity to a much more general setting.
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Strongly transitive actions on Euclidean buildings
We prove a decomposition result for a group $G$ acting strongly transitively on the Tits boundary of a Euclidean building. As an application we provide a local to global result for discrete Euclidean buildings, which generalizes results in the locally compact case by Caprace--Ciobotaru and Burger--Mozes. Let $X$ be a Euclidean building without cone factors. If a group $G$ of automorphisms of $X$ acts strongly transitively on the spherical building at infinity $\partial X$, then the $G$-stabilizer of every affine apartment in $X$ contains all reflections along thick walls. In particular $G$ acts strongly transitively on $X$ if $X$ is simplicial and thick.
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On small abstract quotients of Lie groups and locally compact groups
We prove continuity results for abstract epimorphisms of locally compact groups onto finitely generated groups.
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Transitive actions of locally compact groups on locally contractible spaces
Suppose that $X=G/K$ is the quotient of a locally compact group by a closed subgroup. If $X$ is locally contractible and connected, we prove that $X$ is a manifold. If the $G$-action is faithful, then $G$ is a Lie group.
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Homogeneous compact geometries
We classify compact homogeneous geometries of irreducible spherical type and rank at least 2 which admit a transitive action of a compact connected group, up to equivariant 2-coverings. We apply our classification to polar actions on compact symmetric spaces.
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Metric Properties of Euclidean Buildings
This is a survey on nondiscrete euclidean buildings, with a focus on metric properties of these spaces.
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