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<?xml version="1.0" encoding="utf-8"?> <doi_batch xmlns="http://www.crossref.org/schema/4.3.6" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns:jats="http://www.ncbi.nlm.nih.gov/JATS1" xmlns:ai="http://www.crossref.org/AccessIndicators.xsd" xmlns:mml="http://www.w3.org/1998/Math/MathML" version="4.3.6" xsi:schemaLocation="http://www.crossref.org/schema/4.3.6 https://www.crossref.org/schemas/crossref4.3.6.xsd"> <head> <doi_batch_id>_1596638740</doi_batch_id> <timestamp>1596638740</timestamp> <depositor> <depositor_name>Ukrainskyi Matematychnyi Zhurnal</depositor_name> <email_address>vasyl.ostrovskyi@gmail.com</email_address> </depositor> <registrant>Institute of Mathematics, NAS of Ukraine</registrant> </head> <body> <journal> <journal_metadata> <full_title>Ukrains’kyi Matematychnyi Zhurnal</full_title> <abbrev_title>Ukr. Mat. Zhurn.</abbrev_title> <issn media_type="print">1027-3190</issn> </journal_metadata> <journal_issue> <publication_date media_type="online"> <month>07</month> <day>20</day> <year>2020</year> </publication_date> <journal_volume> <volume>72</volume> </journal_volume> <issue>7</issue> </journal_issue> <journal_article xmlns:jats="http://www.ncbi.nlm.nih.gov/JATS1" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:ai="http://www.crossref.org/AccessIndicators.xsd" publication_type="full_text" metadata_distribution_opts="any"> <titles> <title>Another proof for the continuity of the Lipsman mapping</title> </titles> <contributors> <person_name contributor_role="author" sequence="first"> <given_name>A. </given_name> <surname>Messaoud</surname> </person_name> <person_name contributor_role="author" sequence="additional"> <given_name>A.</given_name> <surname> Rahali</surname> </person_name> </contributors> <jats:abstract xmlns:jats="http://www.ncbi.nlm.nih.gov/JATS1" xmlns:mml="http://www.w3.org/1998/Math/MathML"> <jats:p xmlns:mml="http://www.w3.org/1998/Math/MathML">UDC 515.1 We consider the semidirect product <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>G</mml:mi> <mml:mo>=</mml:mo> <mml:mi>K</mml:mi> <mml:mo>⋉</mml:mo> <mml:mi>V</mml:mi> </mml:mrow> </mml:math> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>K</mml:mi> </mml:mrow> </mml:math> is a connected compact Lie group acting by automorphisms on a finite dimensional real vector space <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>V</mml:mi> </mml:mrow> </mml:math> equipped with an inner product <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo form="prefix">〈</mml:mo> <mml:mo>,</mml:mo> <mml:mo form="postfix">〉</mml:mo> </mml:mrow> </mml:math>. By <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mover accent="true"> <mml:mi>G</mml:mi> <mml:mo>^</mml:mo> </mml:mover> </mml:mrow> </mml:math> we denote the unitary dual of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>G</mml:mi> </mml:mrow> </mml:math> and by <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow> <mml:msup> <mml:mi>𝔤</mml:mi> <mml:mo>‡</mml:mo> </mml:msup> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>G</mml:mi> </mml:mrow> </mml:math> the space of admissible coadjoint orbits, where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>𝔤</mml:mi> </mml:mrow> </mml:math> is the Lie algebra of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>G</mml:mi> </mml:mrow> </mml:math>. It was pointed out by Lipsman that the correspondence between <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mover accent="true"> <mml:mi>G</mml:mi> <mml:mo>^</mml:mo> </mml:mover> </mml:mrow> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow> <mml:msup> <mml:mi>𝔤</mml:mi> <mml:mo>‡</mml:mo> </mml:msup> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>G</mml:mi> </mml:mrow> </mml:math> is bijective. Under some assumption on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>G</mml:mi> </mml:mrow> </mml:math>, we give another proof for the continuity of the orbit mapping (Lipsman mapping)<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mtable class="m-equation-square" displaystyle="true" style="display: block; margin-top: 1.0em; margin-bottom: 2.0em"> <mml:mtr> <mml:mtd> <mml:mspace width="6.0em"/> </mml:mtd> <mml:mtd columnalign="left"> <mml:mi>Θ</mml:mi> <mml:mo>:</mml:mo> <mml:mrow> <mml:msup> <mml:mi>𝔤</mml:mi> <mml:mo>‡</mml:mo> </mml:msup> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>G</mml:mi> <mml:mo>-</mml:mo> <mml:mo>→</mml:mo> <mml:mover accent="true"> <mml:mi>G</mml:mi> <mml:mo>^</mml:mo> </mml:mover> <mml:mo>.</mml:mo> </mml:mtd> </mml:mtr> </mml:mtable> </mml:math></jats:p> </jats:abstract> <publication_date media_type="online"> <month>07</month> <day>15</day> <year>2020</year> </publication_date> <pages> <first_page>945</first_page> <last_page>951</last_page> </pages> <ai:program xmlns:ai="http://www.crossref.org/AccessIndicators.xsd" name="AccessIndicators"> <ai:license_ref>https://creativecommons.org/licenses/by/4.0</ai:license_ref> </ai:program> <doi_data> <doi>10.37863/umzh.v72i7.548</doi> <resource>http://umj.imath.kiev.ua/index.php/umj/article/view/548</resource> <collection property="crawler-based"> <item crawler="iParadigms"> <resource>http://umj.imath.kiev.ua/index.php/umj/article/download/548/8730</resource> </item> </collection> <collection property="text-mining"> <item> <resource mime_type="application/pdf">http://umj.imath.kiev.ua/index.php/umj/article/download/548/8730</resource> </item> </collection> </doi_data> <citation_list> <citation key="21450"> <unstructured_citation>D. Arnal, M. Ben Ammar, M. Selmi, <em>Le problème de la réduction à un sous-groupe dans la quantification par déformation</em>, Ann. Fac. Sci. Toulouse Math. (5), <b>12</b>, 7 – 27 (1991), http://www.numdam.org/item?id=AFST_1991_5_12_1_7_0</unstructured_citation> </citation> <citation key="21451"> <unstructured_citation>W. Baggett, <em>A description of the topology on the dual spaces of certain locally compact groups</em>, Trans. Amer. Math. Soc., <b>132</b>, 175 – 215 (1968), https://doi.org/10.2307/1994889</unstructured_citation> </citation> <citation key="21452"> <unstructured_citation>P. Baguis, <em>Semidirect product and the Pukanszky condition</em>, J. Geom. and Phys., <b>25</b>, 245 – 270 (1998), https://doi.org/10.1016/S0393-0440(97)00028-4</unstructured_citation> </citation> <citation key="21453"> <unstructured_citation>M. Ben Halima, A. Rahali, <em>On the dual topology of a class of Cartan motion groups</em>, J. 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Ludwig, <em>Unitary representation theory of exponential Lie groups</em>, de Gruyter, Berlin (1994), http://www.numdam.org/item?id=ASENS_1973_4_6_4_413_0</unstructured_citation> </citation> <citation key="21458"> <unstructured_citation>R. L. Lipsman, <em>Orbit theory and harmonic analysis on Lie groups with co-compact nilradical</em>, J. Math. Pures et Appl., <b>59</b>, no. 3, 337 – 374 (1980).</unstructured_citation> </citation> <citation key="21459"> <unstructured_citation>A. Rahali, <em>Dual topology of generalized motion groups</em>, Math. Rep., <b>20(70)</b>, no. 3, 233 – 243 (2018).</unstructured_citation> </citation> <citation key="21460"> <unstructured_citation>A. Messaoud, A. Rahali, <em>On the continuity of the Lipsman mapping of semidirect products</em>, Rev. Roum. Math. Pures et Appl., <b>3(63)</b>, 249 – 258 (2018)</unstructured_citation> </citation> </citation_list> </journal_article> </journal> </body> </doi_batch>