A topological criterion for group decompositions

Research output: Contribution to journalArticle

2 Citations (Scopus)

Abstract

Given a connected Lie group G and a closed connected subgroup H of G we prove a necessary and sufficient condition that G decomposes into the Cartesian product of H with G/H is that a similar decomposition holds for the maximal compact subgroups of G and H. Our criterion is applied to the three series of groups for which G/H is SO0(p, q)/SO0(p, q − 1), SU(q+ 1, q + 1)/S[U(q + 1,q) × U/(1)], and SU(q+1, q+1)/SL(n, C)⋊H(n) (p, q ⋟ 1), and we list the values of p and q for which G ≅ H × G/H in each of the three cases. We describe certain decompositions for some of the groups. We show the usefulness of our criterion in obtaining a characterization of the space of differentiable vectors for a unitary induced group representation, and, finally, we show by example of SU(2, 2), how the asymptotic properties of certain function spaces for induced group representations are readily obtained using our results. Our results should be of interest to those working in de Sitter and conformal field theories.

Original languageEnglish (US)
Pages (from-to)285-298
Number of pages14
JournalMathematical Proceedings of the Cambridge Philosophical Society
Volume103
Issue number2
DOIs
StatePublished - Jan 1 1988

Fingerprint

Group Representation
Decompose
Subgroup
Analytic group
Conformal Field Theory
Cartesian product
Function Space
Asymptotic Properties
Differentiable
Necessary Conditions
Closed
Series
Sufficient Conditions

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

Cite this

@article{9cf748098b1b4f5085f6cf19508e00b7,
title = "A topological criterion for group decompositions",
abstract = "Given a connected Lie group G and a closed connected subgroup H of G we prove a necessary and sufficient condition that G decomposes into the Cartesian product of H with G/H is that a similar decomposition holds for the maximal compact subgroups of G and H. Our criterion is applied to the three series of groups for which G/H is SO0(p, q)/SO0(p, q − 1), SU(q+ 1, q + 1)/S[U(q + 1,q) × U/(1)], and SU(q+1, q+1)/SL(n, C)⋊H(n) (p, q ⋟ 1), and we list the values of p and q for which G ≅ H × G/H in each of the three cases. We describe certain decompositions for some of the groups. We show the usefulness of our criterion in obtaining a characterization of the space of differentiable vectors for a unitary induced group representation, and, finally, we show by example of SU(2, 2), how the asymptotic properties of certain function spaces for induced group representations are readily obtained using our results. Our results should be of interest to those working in de Sitter and conformal field theories.",
author = "J. Hebda and Moylan, {Patrick J.}",
year = "1988",
month = "1",
day = "1",
doi = "10.1017/S0305004100064859",
language = "English (US)",
volume = "103",
pages = "285--298",
journal = "Mathematical Proceedings of the Cambridge Philosophical Society",
issn = "0305-0041",
publisher = "Cambridge University Press",
number = "2",

}

A topological criterion for group decompositions. / Hebda, J.; Moylan, Patrick J.

In: Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 103, No. 2, 01.01.1988, p. 285-298.

Research output: Contribution to journalArticle

TY - JOUR

T1 - A topological criterion for group decompositions

AU - Hebda, J.

AU - Moylan, Patrick J.

PY - 1988/1/1

Y1 - 1988/1/1

N2 - Given a connected Lie group G and a closed connected subgroup H of G we prove a necessary and sufficient condition that G decomposes into the Cartesian product of H with G/H is that a similar decomposition holds for the maximal compact subgroups of G and H. Our criterion is applied to the three series of groups for which G/H is SO0(p, q)/SO0(p, q − 1), SU(q+ 1, q + 1)/S[U(q + 1,q) × U/(1)], and SU(q+1, q+1)/SL(n, C)⋊H(n) (p, q ⋟ 1), and we list the values of p and q for which G ≅ H × G/H in each of the three cases. We describe certain decompositions for some of the groups. We show the usefulness of our criterion in obtaining a characterization of the space of differentiable vectors for a unitary induced group representation, and, finally, we show by example of SU(2, 2), how the asymptotic properties of certain function spaces for induced group representations are readily obtained using our results. Our results should be of interest to those working in de Sitter and conformal field theories.

AB - Given a connected Lie group G and a closed connected subgroup H of G we prove a necessary and sufficient condition that G decomposes into the Cartesian product of H with G/H is that a similar decomposition holds for the maximal compact subgroups of G and H. Our criterion is applied to the three series of groups for which G/H is SO0(p, q)/SO0(p, q − 1), SU(q+ 1, q + 1)/S[U(q + 1,q) × U/(1)], and SU(q+1, q+1)/SL(n, C)⋊H(n) (p, q ⋟ 1), and we list the values of p and q for which G ≅ H × G/H in each of the three cases. We describe certain decompositions for some of the groups. We show the usefulness of our criterion in obtaining a characterization of the space of differentiable vectors for a unitary induced group representation, and, finally, we show by example of SU(2, 2), how the asymptotic properties of certain function spaces for induced group representations are readily obtained using our results. Our results should be of interest to those working in de Sitter and conformal field theories.

UR - http://www.scopus.com/inward/record.url?scp=84976015417&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84976015417&partnerID=8YFLogxK

U2 - 10.1017/S0305004100064859

DO - 10.1017/S0305004100064859

M3 - Article

VL - 103

SP - 285

EP - 298

JO - Mathematical Proceedings of the Cambridge Philosophical Society

JF - Mathematical Proceedings of the Cambridge Philosophical Society

SN - 0305-0041

IS - 2

ER -