Search results for "Compact group"
showing 10 items of 26 documents
The probability that $x$ and $y$ commute in a compact group
2010
We show that a compact group $G$ has finite conjugacy classes, i.e., is an FC-group if and only if its center $Z(G)$ is open if and only if its commutator subgroup $G'$ is finite. Let $d(G)$ denote the Haar measure of the set of all pairs $(x,y)$ in $G \times G$ for which $[x,y] = 1$; this, formally, is the probability that two randomly picked elements commute. We prove that $d(G)$ is always rational and that it is positive if and only if $G$ is an extension of an FC-group by a finite group. This entails that $G$ is abelian by finite. The proofs involve measure theory, transformation groups, Lie theory of arbitrary compact groups, and representation theory of compact groups. Examples and re…
The probability that $x^m$ and $y^n$ commute in a compact group
2013
In a recent article [K.H. Hofmann and F.G. Russo, The probability that $x$ and $y$ commute in a compact group, Math. Proc. Cambridge Phil. Soc., to appear] we calculated for a compact group $G$ the probability $d(G)$ that two randomly picked elements $x, y\in G$ satisfy $xy=yx$, and we discussed the remarkable consequences on the structure of $G$ which follow from the assumption that $d(G)$ is positive. In this note we consider two natural numbers $m$ and $n$ and the probabilty $d_{m,n}(G)$ that for two randomly selected elements $x, y\in G$ the relation $x^my^n=y^nx^m$ holds. The situation is more complicated whenever $n,m>1$. If $G$ is a compact Lie group and if its identity component $G_…
Finding Invariants of Group Actions on Function Spaces, a General Methodology from Non-Abelian Harmonic Analysis
2008
In this paper, we describe a general method using the abstract non-Abelian Fourier transform to construct “rich” invariants of group actions on functional spaces.
Henstock type integral in harmonic analysis on zero-dimensional groups
2006
AbstractA Henstock type integral is defined on compact subsets of a locally compact zero-dimensional abelian group. This integral is applied to obtain an inversion formula for the multiplicative integral transform.
Algebraic and Differential Star Products on Regular Orbits of Compact Lie Groups
2000
In this paper we study a family of algebraic deformations of regular coadjoint orbits of compact semisimple Lie groups with the Kirillov Poisson bracket. The deformations are restrictions of deformations on the dual of the Lie algebra. We prove that there are non isomorphic deformations in the family. The star products are not differential, unlike the star products considered in other approaches. We make a comparison with the differential star product canonically defined by Kontsevich's map.
Inversion formulae for the integral transform on a locally compact zero-dimensional group
2009
Abstract Generalized inversion formulae for multiplicative integral transform with a kernel defined by characters of a locally compact zero-dimensional abelian group are obtained using a Kurzweil-Henstock type integral.
Character correspondences in blocks with normal defect groups
2014
Abstract In this paper we give an extension of the Glauberman correspondence to certain characters of blocks with normal defect groups.
THREE-D SINGLETONS AND 2-D C.F.T.
1992
Two-dimensional Wess-Zumino-Novikov-Witten theory is extended to three dimensions, where it becomes a scalar gauge theory of the singleton type. The three-dimensional formulation involves a scalar field valued in a compact group G, a Nakanishi-Lautrup field valued in Lie (G) and Faddeev-Popov ghosts. The physical sector, characterized by the vanishing of the Nakanishi-Lautrup field, coincides with the WZNW theory of the group G. Three-dimensional space-time structure involves a generalized metric, but only its boundary values are of consequence. An alternative formulation in terms of left and right movers (in three dimensions!) is also possible.
XMM-Newton First-Light Observations of the Hickson Galaxy Group 16
2000
This paper presents the XMM-Newton first-light observations of the Hickson-16 compact group of galaxies. Groups are possibly the oldest large-scale structures in the Universe, pre-dating clusters of galaxies, and are highly evolved. This group of small galaxies, at a redshift of 0.0132 (or 80 Mpc) is exceptional in the having the highest concentration of starburst or AGN activity in the nearby Universe. So it is a veritable laboratory for the study of the relationship between galaxy interactions and nuclear activity. Previous optical emission line studies indicated a strong ionising continuum in the galaxies, but its origin, whether from starbursts, or AGN, was unclear. Combined imaging and…
Near abelian profinite groups
2012
Abstract A compact p-group G (p prime) is called near abelian if it contains an abelian normal subgroup A such that G/A has a dense cyclic subgroup and that every closed subgroup of A is normal in G. We relate near abelian groups to a class of compact groups, which are rich in permuting subgroups. A compact group is called quasihamiltonian (or modular) if every pair of compact subgroups commutes setwise. We show that for p ≠ 2 a compact p-group G is near abelian if and only if it is quasihamiltonian. The case p = 2 is discussed separately.