射影表示

在表示论中，群 $G$ 在域 $F$ 上的向量空間 $V$ 上的射影表示指从 $G$ 到射影线性群 $PGL$ 的一個群同態

G\to \mathrm {PGL} (V):=\mathrm {GL} (V)/F^{*}

其中 $\mathrm {GL} (V)$ 表示在域 $F$ 上向量空間 $V$ 的可逆线性变换构成的一般线性群，而 $F^{*}$ 視為純量積映射 $v\mapsto cv$ ，其中 $c\in F^{*}$ 。

若 $V$ 維度有限，選定基底後可將 $\mathrm {PGL} (V)$ 理解為 $\mathrm {PGL} (n,F)$ ，即 $n\times n$ 階可逆矩陣對正規子群 $F^{*}\cdot \mathrm {id} _{V}$ 之商群。

對於給定的群表示 $\rho :G\to \mathrm {GL} (V)$ ，與商映射 $p:\mathrm {GL} (V)\to \mathrm {PGL} (V)$ 合成後可得到一個射影表示。較常探討的是逆向的問題：如何將一個射影表示 ${\bar {\rho }}:G\to \mathrm {PGL} (V)$ 提升至一個表示 $\rho :G\to \mathrm {GL} (V)$ ，使得 $p\circ \rho ={\bar {\rho }}$ ？

對於提升問題，通常採取如下進路：取同態 ${\bar {\rho }}:G\to \mathrm {PGL} (V)$ 與 $p:\mathrm {GL} (V)\to \mathrm {PGL} (V)$ 的纖維積，得到一個中心擴張

1\to F^{*}\to {\tilde {G}}\to \mathrm {GL} (V)\to 1

其中 ${\tilde {G}}=\{(g,M)\in G\times \mathrm {GL} (V)|p(M)=\rho (g)\}$ 。

這類擴張由群上同調 $H^{2}(\mathrm {GL} (V),F^{*})$ 分類。若此擴張是平凡的，則 ${\bar {\rho }}$ 可提升至 $G$ 的表示。即使此表示無法提升，仍可退而求其次，藉群上同調研究擴張的性質，例如：擴張對應的上同調類 $\alpha \in H^{2}(\mathrm {GL} (V),F^{*})$ 滿足 $n\alpha =0$ 若且唯若 ${\bar {\rho }}$ 可提昇為某個中心擴張 $1\to \mathbf {\mu } _{n}\to {\hat {G}}\to \mathrm {GL} (V)\to 1$ 的 ${\hat {G}}$ 的表示。

參見

Read other articles:

Izin Masuk Warga Taiwan

Sampul depan Izin Masuk Warga Taiwan Izin Masuk Warga Taiwan (Hanzi: 台灣居民來往大陸大陸通行證) adalah surat perjalanan yang khusus diberikan oleh pemerintah Republik Rakyat Tiongkok kepada warga Republik Tiongkok. Di sini, walaupun disebutkan bahwa izin hanya diberikan untuk warga yang bertempat tinggal di Taiwan, tetapi secara luas, seluruh warga Republik Tiongkok yang memegang paspor Republik Tiongkok (Taiwan) diharuskan memiliki surat perjalanan ini untuk memasuki wilayah Re...

Kereta rel listrik JR East seri 205

Artikel ini bukan mengenai CC205.KRL JR East seri 205KRL seri 205 rangkaian DP10 (eks KeYo M17) dengan livery baru KAI CommuterBeroperasiYaPembuat Kawasaki Heavy Industries Nippon Sharyo Hitachi, Ltd. Kinki Sharyo Tokyu Car CorporationJR East Ofuna WorksDigantikan olehJR East E231-500 (Jalur Yamanote)JR East E233-6000 (Jalur Yokohama)JR East E233-7000 (Jalur Saikyo)JR East E233-8000 (Jalur Nambu)JR East 209-500, JR East E231-0 (Jalur Musashino)Tahun pembuatan 1984-1991 (rangkaian 4 pintu) 19...

جاكلين بارتون

جاكلين بارتون (بالإنجليزية: Jacqueline Kapelman Barton)‏ معلومات شخصية الميلاد 7 مايو 1952 (72 سنة) نيويورك مواطنة الولايات المتحدة عضوة في الأكاديمية الوطنية للعلوم[1]، والأكاديمية الأمريكية للفنون والعلوم، والجمعية الأمريكية للفلسفة الحياة العملية المد�...

Desmond Tutu

The Most ReverendDesmond TutuUskup Agung Emeritus Cape Town Uskup Desmond Tutu, peraih Nobel Perdamaian tahun 1984Provinsi gerejawiGereja Anglikan Afrika SelatanTakhtaCape Town (pensiun)Awal masa jabatan7 September 1986Masa jabatan berakhir1996PendahuluP.W.R. RussellPenerusNjongonkulu NdunganeJabatan lainUskup LesothoUskup Agung JohannesburgUskup Agung Cape TownImamatTahbisan imamImam Diakon – 1960Tahbisan uskup1976Informasi pribadiNama lahirDesmond Mpilo TutuLahir(1931-10-07)7 Oktober 1931...

Синелобый амазон

Синелобый амазон Научная классификация Домен:ЭукариотыЦарство:ЖивотныеПодцарство:ЭуметазоиБез ранга:Двусторонне-симметричныеБез ранга:ВторичноротыеТип:ХордовыеПодтип:ПозвоночныеИнфратип:ЧелюстноротыеНадкласс:ЧетвероногиеКлада:АмниотыКлада:ЗавропсидыКласс:Пт�...

Синелобый амазон

Juventus Football Club

Pour les articles homonymes, voir JFC. Juventus FC Généralités Nom complet Juventus Turin Football Club S.p.A. Surnoms La Vecchia Signora[1] (La Vieille Dame) I Bianconeri [2](Les Blanc et Noir)La Gobba (La Bossue)Zebre[3] Noms précédents Sport Club Juventus (1897-1900) Foot-Ball Club Juventus (1900-1936) Juventus (1936-1942) Juventus-Cisitalia (1942-1945) Fondation 1er novembre 1897 (126 ans) Statut professionnel Depuis 1929 Couleurs Blanc et Noir Stade Allianz Stadium (41 50...

Mustafa Al-Kadhimi

Mustafa al-Kadhimiمصطفى الكاظميAl-Kadhimi pada Mei 2020 Perdana Menteri Irak ke-50Masa jabatan6 Mei 2020 – 28 Oktober 2022PresidenBarham SalihPendahuluAdil Abdul-MahdiPenggantiPetahanaMenteri Urusan Luar NegeriMasa jabatan12 Mei 2020 – 6 Juni 2020PendahuluMohamed Ali AlhakimPenggantiFuad HusseinDirektur INISMasa jabatan7 Juni 2016 – 9 April 2020PresidenFuad Masum Barham SalihPerdana MenteriHaider al-Abadi Adil Abdul-MahdiPendahuluZuheir Fadel Abbas...

Слоёное тесто

Слоёное тесто Страна происхождения Франция Компоненты Основные масло сливочноеводамука Медиафайлы на Викискладе Слоёное тесто — тесто, используемое при выпечке слоёных пирожков, булочек, самсы, других кулинарных изделий. Характерной особенностью слоёного т�...

土库曼斯坦总统

土库曼斯坦总统土库曼斯坦国徽土库曼斯坦总统旗現任谢尔达尔·别尔德穆哈梅多夫自2022年3月19日官邸阿什哈巴德总统府（Oguzkhan Presidential Palace）機關所在地阿什哈巴德任命者直接选举任期7年，可连选连任首任萨帕尔穆拉特·尼亚佐夫设立1991年10月27日土库曼斯坦土库曼斯坦政府与政治国家政府土库曼斯坦宪法国旗国徽国歌立法機關（英语：National Council of Turkmenistan） ...

土库曼斯坦总统

Kazuhiko Yamaguchi

Kazuhiko YamaguchiLahir5 Februari 1937 (umur 87)Prefektur NaganoPekerjaanSutradara film Kazuhiko Yamaguchi (山口和彦code: ja is deprecated , Yamaguchi Kazuhiko, lahir 5 Februari 1937) adalah seorang sutradara film Jepang.[1] Karier Kazuhiko lahir di Prefektur Nagano dan lulus dari Universitas Waseda. Ia mulai bekerja di studio Tōei di Kyoto.[2] Ia menyutradari sejumlah seri film laga pada tahun 1970-an dan juga beberapa film televisi.[2] Filmografi Delinquen...

Economic depression

Sustained, long-term downturn in economic activity in one or more economies The examples and perspective in this article deal primarily with the English-speaking world and do not represent a worldwide view of the subject. You may improve this article, discuss the issue on the talk page, or create a new article, as appropriate. (October 2012) (Learn how and when to remove this message) An economic depression is a period of carried long-term economic downturn that is the result of lowered econo...

Bluestreak cleaner wrasse

Species of fish Bluestreak cleaner wrasse Conservation status Least Concern (IUCN 3.1)[1] Scientific classification Domain: Eukaryota Kingdom: Animalia Phylum: Chordata Class: Actinopterygii Order: Labriformes Family: Labridae Genus: Labroides Species: L. dimidiatus Binomial name Labroides dimidiatus(Valenciennes, 1839) Synonyms[2] Cossyphus dimidiatus Valenciennes, 1839 Labroides paradiseus Bleeker, 1851 Callyodon ikan Montrouzier, 1857 Labroides bicincta Saville-K...

Henry Taylor (dramatist)

English playwright and poet SirHenry TaylorKCMGSir Henry Taylor, by W. J. Hawker.Born(1800-10-18)18 October 1800Bishop Middleham, EnglandDied27 March 1886(1886-03-27) (aged 85)OccupationDramatist and poetLanguageEnglishSpouseThe Honorable Theodosia Alice Spring RiceChildrenFive: including Ida Alice Ashworth Taylor Sir Henry Taylor KCMG (18 October 1800 – 27 March 1886) was an English dramatist and poet, Colonial Office official, and man of letters. Early life Henry Taylor was born ...

Bosnia and Herzegovina at the 2020 Summer Paralympics

Sporting event delegationBosnia and Herzegovina at the2020 Summer ParalympicsFlag of Bosnia and HerzegovinaIPC codeBIHNPCParalympic Committee of Bosnia and Herzegovinain Tokyo, JapanAugust 24, 2021 (2021-08-24) – September 5, 2021 (2021-09-05)Competitors16 in 4 sportsMedals Gold 0 Silver 0 Bronze 1 Total 1 Summer Paralympics appearances (overview)19962000200420082012201620202024Other related appearances Yugoslavia (1972–2000) Bosnia and Herzegovina ...

2011 royal tour of Canada

Prince William and Catherine, the Duke and Duchess of Cambridge, during Canada Day celebrations at Parliament Hill on 1 July 2011 The 2011 Royal Tour of Canada was undertaken by Prince William, and Catherine, the Duke and Duchess of Cambridge, between 30 June and 8 July 2011. The tour saw the newlywed couple visit all of Canada's regions. It was the first such tour undertaken by the Duke and Duchess since their marriage two months prior,[1] and the first duties the couple carried out...

Judah ha-Nasi

2nd-century rabbi and editor of the Mishnah RabbiJudah ha-NasiThe Burial Cave of Rabbi Yehuda HaNasi in Beit Shearim, IsraelBornYehudah ben Shimonc. 135 CEJudea, Roman EmpireDiedc. 217 CESepphoris, Roman JudeaOther namesRabbi, Rebbi, Rabbenu HaQadoshOccupation(s)Rabbi, Tanna, Chief redactor of the MishnahKnown forCompilation of the Mishnah, leadership of the Jewish community in Roman JudeaNotable workMishnah Rebbi redirects here. For the title, see Honorifics in Judaism. For a Torah...

Giovanni Paschetta

Questa voce sull'argomento calciatori italiani è solo un abbozzo. Contribuisci a migliorarla secondo le convenzioni di Wikipedia. Segui i suggerimenti del progetto di riferimento. Giovanni PaschettaNazionalità Italia Altezza185 cm Peso75 kg Calcio RuoloAllenatore (ex difensore) Termine carriera2011 - giocatore CarrieraGiovanili 19??-1993 Cosenza Squadre di club1 1993-1997 Cosenza38 (1)1997 Atletico Catania13 (0)1997-1999 Cosenza56 (1)1999-2000 Crotone22 (...

Kicking & Screaming (Sebastian Bach album)

2011 studio album by Sebastian BachKicking & ScreamingStudio album by Sebastian BachReleasedSeptember 21, 2011 (Japan)September 23, 2011 (Europe)September 27, 2011 (US)RecordedFebruary–May 2011StudioPrecision Mastering Studio (Los Angeles)GenreHeavy metal, hard rockLength51:18LabelFrontiersGet Off My Bach!AvalonProducerBob MarletteSebastian Bach chronology Angel Down(2007) Kicking & Screaming(2011) Give 'Em Hell(2014) Singles from Kicking & Screaming Kicking and Screami...