Rondedzera Lie Algebra
Context:Mumasvomhu, boka reLie rakapfava iboka reLie rakabatana risiri reAbelian G risina mapoka epakati akabatana. Rondedzero yemapoka akareruka eLie anogona kushandiswa kuverenga kubva pane runyorwa rweakareruka eLie algebras uye Riemannian symmetric nzvimbo.  Pamwe chete neboka reLie rinoshandurwa renhamba chaidzo, \mathbb {R}, uye iyo yeyuniti-hukuru nhamba dzakaomarara, U(1) (denderedzwa yeyuniti), mapoka akareruka eLie anopa maatomu "mabhuroko" anoumba ese (akaenzana-akaenzana) akabatana mapoka eLie kuburikidza nekushanda kweboka rekuwedzera. Mapoka mazhinji anowanzosangana neLie ari nyore kana kuti 'pedyo' nekuva nyore: somuenzaniso, inonzi "boka rinokosha rakatwasuka" SL ((n) ye n ne n matrix ne determinant yakaenzana ne 1 iri nyore kune vose n > 1.  Kugoverwa kwokutanga kwamapoka akareruka aLie kwakaitwa naWilhelm Killing, uye basa iri rakazogadziridzwa naÉlie Cartan. Kurongwa kwokupedzisira kunowanzonzi kurongwa kwaKilling-Cartan.
Zvinosuruvarisa kuti hapana tsanangudzo inogamuchirwa pasi pose yeboka raLie riri nyore. Zvikurukuru, haisi nguva dzose inotsanangurwa seboka reLie iro riri nyore seboka risinganzwisisiki. Vanyori vanosiyana pamusoro pekuti boka reLie rakapfava rinofanira kubatanidzwa, kana kuti rinobvumirwa kuva nechepakati chisina kukosha, kana kuti kana  \mathbb {R} iboka raLie rakapfava.  Tsananguro yakajairika ndeyekuti boka reLie rakareruka kana rakabatana, risiri raAbelian, uye rimwe nerimwe rakavharwa rakabatana rakajairika subgroup ingave kuzivikanwa kana boka rose. Zvikurukuru, mapoka akareruka anobvumirwa kuva nechepakati chisiri chiduku, asi  hakusi nyore.  Munyaya ino mapoka akareruka aLie akabatanidzwa ane nzvimbo isina basa akanyorwa. Kana izvi zvazivikanwa, izvo zvine nzvimbo isiri duku zviri nyore kunyora sezvinotevera. Chero boka reLie riri nyore rine pakati risingakoshi rine chifukidziro chepasi pose, chine pakati riri boka rinokosha reboka reLie riri nyore. Mapoka akareruka aLie anowirirana ane nzvimbo isiri duku anogona kuwanwa sezvibatanidzwa zvechifukidziro ichi chepasi pose neboka duku repakati.  Dzimwe nzira Tsananguro yakaenzana yeboka reLie iri nyore inotevera kubva pakunyorerana kweLie: Boka reLie rakabatana riri nyore kana Lie algebra yaro iri nyore. Chinhu chinokosha chezvokugadzira ndechokuti boka reLie rinogona kuva nemapoka maduku akaenzana. Nokuda kwechikonzero ichi, tsanangudzo yeboka reLie iri nyore haisi yakaenzana nekutsanangurwa kweboka reLie iro riri nyore seboka risinganzwisisiki.  Mapoka akareruka aLie anosanganisira mapoka akawanda ekare aLie, anopa rutsigiro rwezvemapoka e spherical geometry, projective geometry uye geometries dzakabatana mupfungwa yechirongwa chaFelix Klein's Erlangen. Zvakabuda mukufamba kwekurongwa kwemapoka akareruka aLie kuti kunewo mikana inoverengeka inoshamisa isingawirirane nechero geometry inozivikanwa. Aya mapoka akasarudzika anogovera mienzaniso yakawanda chaiyo nemagadzirirwo mune mamwe matavi emasvomhu, pamwe chete nephysics yemazuva ano.  Somuenzaniso unopesana, boka romutsara rose harisi nyore, kana kuti risiri nyore. Ikoku imhaka yokuti zviverengero zvechizivikanwe zvinoumba boka duku risiri renguva dzose, nokudaro richidzivisa rondedzero yacho. Nenzira yakaenzana, Lie algebra yakaenzana ine chimiro cheKilling chakashata, nokuti zvizhinji zvechivimbiso chemepu kune zero element yealgebra. Nokudaro, Lie algebra inowiriranawo haisi nyore kana kuti semisimple. Mumwe muenzaniso unopesana nemapoka chaiwo eorthogonal mune kunyange ukuru. Izvi zvine matrix -  -I iri pakati, uye chinhu ichi chakabatana nenzira nechinhu chinozivikanwa, uye saka mapoka aya anonzvenga tsananguro. Ose ari maviri aya mapoka anoderedza.