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Search: MSC category 22 ( Topological groups, Lie groups )

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1. CMB Online first

Jensen, Gerd; Pommerenke, Christian
On the structure of the Schild group in Relativity Theory
Alfred Schild has established conditions that Lorentz transformations map world-vectors $(ct,x,y,z)$ with integer coordinates onto vectors of the same kind. These transformations are called integral Lorentz transformations. The present paper contains supplements to our earlier work with a new focus on group theory. To relate the results to the familiar matrix group nomenclature we associate Lorentz transformations with matrices in $\mathrm{SL}(2,\mathbb{C})$. We consider the lattice of subgroups of the group originated in Schild's paper and obtain generating sets for the full group and its subgroups.

Keywords:Lorentz transformation, integer lattice, Gaussian integers, Schild group, subgroup
Categories:22E43, 20H99, 83A05

2. CMB Online first

Christ, Michael; Rieffel, Marc A.
Nilpotent group C*-algebras as compact quantum metric spaces
Let $\mathbb{L}$ be a length function on a group $G$, and let $M_\mathbb{L}$ denote the operator of pointwise multiplication by $\mathbb{L}$ on $\lt(G)$. Following Connes, $M_\mathbb{L}$ can be used as a ``Dirac'' operator for the reduced group C*-algebra $C_r^*(G)$. It defines a Lipschitz seminorm on $C_r^*(G)$, which defines a metric on the state space of $C_r^*(G)$. We show that for any length function satisfying a strong form of polynomial growth on a discrete group, the topology from this metric coincides with the weak-$*$ topology (a key property for the definition of a ``compact quantum metric space''). In particular, this holds for all word-length functions on finitely generated nilpotent-by-finite groups.

Keywords:group C*-algebra, Dirac operator, quantum metric space, discrete nilpotent group, polynomial growth
Categories:46L87, 20F65, 22D15, 53C23, 58B34

3. CMB 2016 (vol 59 pp. 244)

Cao, Wensheng; Huang, Xiaolin
A Note on Quaternionic Hyperbolic Ideal Triangle Groups
In this paper, the quaternionic hyperbolic ideal triangle groups are parameterized by a real one-parameter family $\{\phi_s: s\in \mathbb{R}\}$. The indexing parameter $s$ is the tangent of the quaternionic angular invariant of a triple of points in $\partial \mathbf{H}_{\mathbb{h}}^2 $ forming this ideal triangle. We show that if $s \gt \sqrt{125/3}$ then $\phi_s$ is not a discrete embedding, and if $s \leq \sqrt{35}$ then $\phi_s$ is a discrete embedding.

Keywords:quaternionic inversion, ideal triangle group, quaternionic Cartan angular invariant
Categories:20F67, 22E40, 30F40

4. CMB 2015 (vol 59 pp. 123)

Jensen, Gerd; Pommerenke, Christian
Discrete Space-time and Lorentz Transformations
Alfred Schild has established conditions that Lorentz transformations map world-vectors $(ct,x,y,z)$ with integer coordinates onto vectors of the same kind. The problem was dealt with in the context of tensor and spinor calculus. Due to Schild's number-theoretic arguments, the subject is also interesting when isolated from its physical background. The paper of Schild is not easy to understand. Therefore we first present a streamlined version of his proof which is based on the use of null vectors. Then we present a purely algebraic proof that is somewhat shorter. Both proofs rely on the properties of Gaussian integers.

Keywords:Lorentz transformation, integer lattice, Gaussian integers
Categories:22E43, 20H99, 83A05

5. CMB 2015 (vol 58 pp. 632)

Silberman, Lior
Quantum Unique Ergodicity on Locally Symmetric Spaces: the Degenerate Lift
Given a measure $\bar\mu_\infty$ on a locally symmetric space $Y=\Gamma\backslash G/K$, obtained as a weak-{*} limit of probability measures associated to eigenfunctions of the ring of invariant differential operators, we construct a measure $\bar\mu_\infty$ on the homogeneous space $X=\Gamma\backslash G$ which lifts $\bar\mu_\infty$ and which is invariant by a connected subgroup $A_{1}\subset A$ of positive dimension, where $G=NAK$ is an Iwasawa decomposition. If the functions are, in addition, eigenfunctions of the Hecke operators, then $\bar\mu_\infty$ is also the limit of measures associated to Hecke eigenfunctions on $X$. This generalizes results of the author with A. Venkatesh in the case where the spectral parameters stay away from the walls of the Weyl chamber.

Keywords:quantum unique ergodicity, microlocal lift, spherical dual
Categories:22E50, 43A85

6. CMB 2013 (vol 57 pp. 357)

Lauret, Emilio A.
Representation Equivalent Bieberbach Groups and Strongly Isospectral Flat Manifolds
Let $\Gamma_1$ and $\Gamma_2$ be Bieberbach groups contained in the full isometry group $G$ of $\mathbb{R}^n$. We prove that if the compact flat manifolds $\Gamma_1\backslash\mathbb{R}^n$ and $\Gamma_2\backslash\mathbb{R}^n$ are strongly isospectral then the Bieberbach groups $\Gamma_1$ and $\Gamma_2$ are representation equivalent, that is, the right regular representations $L^2(\Gamma_1\backslash G)$ and $L^2(\Gamma_2\backslash G)$ are unitarily equivalent.

Keywords:representation equivalent, strongly isospectrality, compact flat manifolds
Categories:58J53, 22D10

7. CMB 2012 (vol 56 pp. 881)

Xie, BaoHua; Wang, JieYan; Jiang, YuePing
Free Groups Generated by Two Heisenberg Translations
In this paper, we will discuss the groups generated by two Heisenberg translations of $\mathbf{PU}(2,1)$ and determine when they are free.

Keywords:free group, Heisenberg group, complex triangle group
Categories:30F40, 22E40, 20H10

8. CMB 2012 (vol 57 pp. 424)

Sołtan, Piotr M.; Viselter, Ami
A Note on Amenability of Locally Compact Quantum Groups
In this short note we introduce a notion called ``quantum injectivity'' of locally compact quantum groups, and prove that it is equivalent to amenability of the dual. Particularly, this provides a new characterization of amenability of locally compact groups.

Keywords:amenability, conditional expectation, injectivity, locally compact quantum group, quantum injectivity
Categories:20G42, 22D25, 46L89

9. CMB 2012 (vol 56 pp. 709)

Bartošová, Dana
Universal Minimal Flows of Groups of Automorphisms of Uncountable Structures
It is a well-known fact, that the greatest ambit for a topological group $G$ is the Samuel compactification of $G$ with respect to the right uniformity on $G.$ We apply the original description by Samuel from 1948 to give a simple computation of the universal minimal flow for groups of automorphisms of uncountable structures using Fraïssé theory and Ramsey theory. This work generalizes some of the known results about countable structures.

Keywords:universal minimal flows, ultrafilter flows, Ramsey theory
Categories:37B05, 03E02, 05D10, 22F50, 54H20

10. CMB 2012 (vol 56 pp. 647)

Valverde, Cesar
On Induced Representations Distinguished by Orthogonal Groups
Let $F$ be a local non-archimedean field of characteristic zero. We prove that a representation of $GL(n,F)$ obtained from irreducible parabolic induction of supercuspidal representations is distinguished by an orthogonal group only if the inducing data is distinguished by appropriate orthogonal groups. As a corollary, we get that an irreducible representation induced from supercuspidals that is distinguished by an orthogonal group is metic.

Keywords:distinguished representation, parabolic induction
Category:22E50

11. CMB 2011 (vol 56 pp. 442)

Zelenyuk, Yevhen
Closed Left Ideal Decompositions of $U(G)$
Let $G$ be an infinite discrete group and let $\beta G$ be the Stone--Čech compactification of $G$. We take the points of $ėta G$ to be the ultrafilters on $G$, identifying the principal ultrafilters with the points of $G$. The set $U(G)$ of uniform ultrafilters on $G$ is a closed two-sided ideal of $\beta G$. For every $p\in U(G)$, define $I_p\subseteq\beta G$ by $I_p=\bigcap_{A\in p}\operatorname{cl} (GU(A))$, where $U(A)=\{p\in U(G):A\in p\}$. We show that if $|G|$ is a regular cardinal, then $\{I_p:p\in U(G)\}$ is the finest decomposition of $U(G)$ into closed left ideals of $\beta G$ such that the corresponding quotient space of $U(G)$ is Hausdorff.

Keywords:Stone--Čech compactification, uniform ultrafilter, closed left ideal, decomposition
Categories:22A15, 54H20, 22A30, 54D80

12. CMB 2011 (vol 56 pp. 213)

Tausk, Daniel V.
A Locally Compact Non Divisible Abelian Group Whose Character Group Is Torsion Free and Divisible
It was claimed by Halmos in 1944 that if $G$ is a Hausdorff locally compact topological abelian group and if the character group of $G$ is torsion free, then $G$ is divisible. We prove that such a claim is false by presenting a family of counterexamples. While other counterexamples are known, we also present a family of stronger counterexamples, showing that even if one assumes that the character group of $G$ is both torsion free and divisible, it does not follow that $G$ is divisible.

Category:22B05

13. CMB 2011 (vol 55 pp. 870)

Wang, Hui; Deng, Shaoqiang
Left Invariant Einstein-Randers Metrics on Compact Lie Groups
In this paper we study left invariant Einstein-Randers metrics on compact Lie groups. First, we give a method to construct left invariant non-Riemannian Einstein-Randers metrics on a compact Lie group, using the Zermelo navigation data. Then we prove that this gives a complete classification of left invariant Einstein-Randers metrics on compact simple Lie groups with the underlying Riemannian metric naturally reductive. Further, we completely determine the identity component of the group of isometries for this type of metrics on simple groups. Finally, we study some geometric properties of such metrics. In particular, we give the formulae of geodesics and flag curvature of such metrics.

Keywords:Einstein-Randers metric, compact Lie groups, geodesic, flag curvature
Categories:17B20, 22E46, 53C12

14. CMB 2011 (vol 56 pp. 218)

Yang, Dilian
Functional Equations and Fourier Analysis
By exploring the relations among functional equations, harmonic analysis and representation theory, we give a unified and very accessible approach to solve three important functional equations - the d'Alembert equation, the Wilson equation, and the d'Alembert long equation - on compact groups.

Keywords:functional equations, Fourier analysis, representation of compact groups
Categories:39B52, 22C05, 43A30

15. CMB 2011 (vol 56 pp. 116)

Krepski, Derek
Central Extensions of Loop Groups and Obstruction to Pre-Quantization
An explicit construction of a pre-quantum line bundle for the moduli space of flat $G$-bundles over a Riemann surface is given, where $G$ is any non-simply connected compact simple Lie group. This work helps to explain a curious coincidence previously observed between Toledano Laredo's work classifying central extensions of loop groups $LG$ and the author's previous work on the obstruction to pre-quantization of the moduli space of flat $G$-bundles.

Keywords:loop group, central extension, prequantization
Categories:53D, 22E

16. CMB 2011 (vol 55 pp. 297)

Glasner, Eli
The Group $\operatorname{Aut}(\mu)$ is Roelcke Precompact
Following a similar result of Uspenskij on the unitary group of a separable Hilbert space, we show that, with respect to the lower (or Roelcke) uniform structure, the Polish group $G= \operatorname{Aut}(\mu)$ of automorphisms of an atomless standard Borel probability space $(X,\mu)$ is precompact. We identify the corresponding compactification as the space of Markov operators on $L_2(\mu)$ and deduce that the algebra of right and left uniformly continuous functions, the algebra of weakly almost periodic functions, and the algebra of Hilbert functions on $G$, i.e., functions on $G$ arising from unitary representations, all coincide. Again following Uspenskij, we also conclude that $G$ is totally minimal.

Keywords:Roelcke precompact, unitary group, measure preserving transformations, Markov operators, weakly almost periodic functions
Categories:54H11, 22A05, 37B05, 54H20

17. CMB 2011 (vol 54 pp. 663)

Haas, Ruth; G. Helminck, Aloysius
Admissible Sequences for Twisted Involutions in Weyl Groups
Let $W$ be a Weyl group, $\Sigma$ a set of simple reflections in $W$ related to a basis $\Delta$ for the root system $\Phi$ associated with $W$ and $\theta$ an involution such that $\theta(\Delta) = \Delta$. We show that the set of $\theta$-twisted involutions in $W$, $\mathcal{I}_{\theta} = \{w\in W \mid \theta(w) = w^{-1}\}$ is in one to one correspondence with the set of regular involutions $\mathcal{I}_{\operatorname{Id}}$. The elements of $\mathcal{I}_{\theta}$ are characterized by sequences in $\Sigma$ which induce an ordering called the Richardson-Springer Poset. In particular, for $\Phi$ irreducible, the ascending Richardson-Springer Poset of $\mathcal{I}_{\theta}$, for nontrivial $\theta$ is identical to the descending Richardson-Springer Poset of $\mathcal{I}_{\operatorname{Id}}$.

Categories:20G15, 20G20, 22E15, 22E46, 43A85

18. CMB 2010 (vol 54 pp. 44)

Cheung, Wai-Shun; Tam, Tin-Yau
Star-Shapedness and $K$-Orbits in Complex Semisimple Lie Algebras
Given a complex semisimple Lie algebra $\mathfrak{g}=\mathfrak{k}+i\mathfrak{k}$ ($\mathfrak{k}$ is a compact real form of $\mathfrak{g}$), let $\pi\colon\mathfrak{g}\to \mathfrak{h}$ be the orthogonal projection (with respect to the Killing form) onto the Cartan subalgebra $\mathfrak{h}:=\mathfrak{t}+i\mathfrak{t}$, where $\mathfrak{t}$ is a maximal abelian subalgebra of $\mathfrak{k}$. Given $x\in \mathfrak{g}$, we consider $\pi(\mathop{\textrm{Ad}}(K) x)$, where $K$ is the analytic subgroup $G$ corresponding to $\mathfrak{k}$, and show that it is star-shaped. The result extends a result of Tsing. We also consider the generalized numerical range $f(\mathop{\textrm{Ad}}(K)x)$, where $f$ is a linear functional on $\mathfrak{g}$. We establish the star-shapedness of $f(\mathop{\textrm{Ad}}(K)x)$ for simple Lie algebras of type $B$.

Categories:22E10, 17B20

19. CMB 2010 (vol 54 pp. 126)

Jin, Yongyang; Zhang, Genkai
Fundamental Solutions of Kohn Sub-Laplacians on Anisotropic Heisenberg Groups and H-type Groups
We prove that the fundamental solutions of Kohn sub-Laplacians $\Delta + i\alpha \partial_t$ on the anisotropic Heisenberg groups are tempered distributions and have meromorphic continuation in $\alpha$ with simple poles. We compute the residues and find the partial fundamental solutions at the poles. We also find formulas for the fundamental solutions for some matrix-valued Kohn type sub-Laplacians on H-type groups.

Categories:22E30, 35R03, 43A80

20. CMB 2008 (vol 51 pp. 60)

Janzen, David
F{\o}lner Nets for Semidirect Products of Amenable Groups
For unimodular semidirect products of locally compact amenable groups $N$ and $H$, we show that one can always construct a F{\o}lner net of the form $(A_\alpha \times B_\beta)$ for $G$, where $(A_\alpha)$ is a strong form of F{\o}lner net for $N$ and $(B_\beta)$ is any F{\o}lner net for $H$. Applications to the Heisenberg and Euclidean motion groups are provided.

Categories:22D05, 43A07, 22D15, 43A20

21. CMB 2007 (vol 50 pp. 632)

Zelenyuk, Yevhen; Zelenyuk, Yuliya
Transformations and Colorings of Groups
Let $G$ be a compact topological group and let $f\colon G\to G$ be a continuous transformation of $G$. Define $f^*\colon G\to G$ by $f^*(x)=f(x^{-1})x$ and let $\mu=\mu_G$ be Haar measure on $G$. Assume that $H=\Imag f^*$ is a subgroup of $G$ and for every measurable $C\subseteq H$, $\mu_G((f^*)^{-1}(C))=\mu_H(C)$. Then for every measurable $C\subseteq G$, there exist $S\subseteq C$ and $g\in G$ such that $f(Sg^{-1})\subseteq Cg^{-1}$ and $\mu(S)\ge(\mu(C))^2$.

Keywords:compact topological group, continuous transformation, endomorphism, Ramsey theoryinversion,
Categories:05D10, 20D60, 22A10

22. CMB 2007 (vol 50 pp. 440)

Raghuram, A.
A Künneth Theorem for $p$-Adic Groups
Let $G_1$ and $G_2$ be $p$-adic groups. We describe a decomposition of ${\rm Ext}$-groups in the category of smooth representations of $G_1 \times G_2$ in terms of ${\rm Ext}$-groups for $G_1$ and $G_2$. We comment on ${\rm Ext}^1_G(\pi,\pi)$ for a supercuspidal representation $\pi$ of a $p$-adic group $G$. We also consider an example of identifying the class, in a suitable ${\rm Ext}^1$, of a Jacquet module of certain representations of $p$-adic ${\rm GL}_{2n}$.

Categories:22E50, 18G15, 55U25

23. CMB 2007 (vol 50 pp. 460)

Spielberg, Jack
Weak Semiprojectivity for Purely Infinite $C^*$-Algebras
We prove that a separable, nuclear, purely infinite, simple $C^*$-algebra satisfying the universal coefficient theorem is weakly semiprojective if and only if its $K$-groups are direct sums of cyclic groups.

Keywords:Kirchberg algebra, weak semiprojectivity, graph $C^*$-algebra
Categories:46L05, 46L80, 22A22

24. CMB 2007 (vol 50 pp. 291)

Sarkar, Rudra P.; Sengupta, Jyoti
Beurling's Theorem and Characterization of Heat Kernel for Riemannian Symmetric Spaces of Noncompact Type
We prove Beurling's theorem for rank $1$ Riemannian symmetric spaces and relate its consequences with the characterization of the heat kernel of the symmetric space.

Keywords:Beurling's Theorem, Riemannian symmetric spaces, uncertainty principle
Categories:22E30, 43A85

25. CMB 2007 (vol 50 pp. 48)

Dvorsky, Alexander
Tensor Square of the Minimal Representation of $O(p,q)$
In this paper, we study the tensor product $\pi=\sigma^{\min}\otimes \sigma^{\min}$ of the minimal representation $\sigma^{\min}$ of $O(p,q)$ with itself, and decompose $\pi$ into a direct integral of irreducible representations. The decomposition is given in terms of the Plancherel measure on a certain real hyperbolic space.

Category:22e46
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