Chaza i - Lie Algebra
Context:Ezemathematics, iqembu elilula likaLie liyiqembu elixhunyiwe elingeyona i-Abelian Lie G elingenayo amaqembu amancane axhunyiwe angavamile. Uhlu lwamaqembu alula ka-Lie lungasetshenziselwa ukufunda uhlu lwama-algebra alula ka-Lie kanye nezikhala ze-Riemannian ezilinganayo.  Ngokubambisana neqembu le-Lie eliguquguqukayo lezinombolo zangempela, \mathbb {R}, kanye nalezo zenombolo eziyinkimbinkimbi ezinesisindo esingu-unit, U(1) (isiyingi seyunithi), amaqembu alula kaLie anikeza "amabhulokhi" e-athomu akha wonke amaqembu kaLie axhunyiwe (anobukhulu obugcwele) ngokusebenza kokunwetshwa kweqembu. Amaqembu amaningi e-Lie avame ukuhlangana nawo alula noma 'asondele' ekubeni alula: ngokwesibonelo, okuthiwa "iqembu eliqondile elikhethekile" SL ((n) le-n nge-n matrices ene-determinant elingana no-1 lilula kuwo wonke ama-n > 1.  Ukuhlukaniswa kokuqala kwamaqembu alula kaLie kwakunguWilhelm Killing, futhi lo msebenzi kamuva wapheleliswa ngu-Élie Cartan. Ukuhlukaniswa kokugcina kuvame ukubizwa ngokuthi ukuhlukaniswa kweKilling-Cartan.
Ngeshwa, ayikho incazelo eyamukelwa emhlabeni wonke yeqembu elilula likaLie. Ikakhulukazi, akuyona njalo ichazwa njengeqembu Lie ukuthi ilula njengoba iqembu abstract. Ababhali abavumelani ngokuthi iqembu elilula likaLie kufanele lihlanganiswe, noma ukuthi livunyelwe ukuba nesikhungo esingeyona into encane, noma ukuthi  \mathbb {R} yiqembu elilula likaLie.  Incazelo evame kakhulu ukuthi iqembu likaLie lilula uma lixhunyiwe, hhayi i-Abelian, futhi wonke amaqembu amancane avulekile axhunyiwe noma yi-identity noma iqembu lonke. Ikakhulukazi, amaqembu alula avunyelwe ukuba nesikhungo esingabalulekile, kodwa  akulula.  Kulesi sihloko amaqembu alula kaLie axhunyiwe anezikhungo ezincane ahlelwe. Uma sekwaziwa, okunezikhungo ezingabalulekile kulula ukukubhala ngalendlela elandelayo. Noma iliphi iqembu elilula le-Lie elinesikhungo esincane linesembozo somhlaba wonke, isikhungo salo okuyiqembu eliyisisekelo leqembu elilula le-Lie. Amaqembu alula ka-Lie ahambelana nesikhungo esingabalulekile angatholakala njengama-quotients alokhu kufakwa kwendawo yonke ngeqembu elincane lesikhungo.  Ezinye izindlela Incazelo elinganayo yeqembu elilula likaLie ilandela ekuhambelaneni kukaLie: Iqembu elixhunyiwe likaLie lilula uma i- Lie algebra yalo ilula. Iphuzu elibalulekile lobuchwepheshe ukuthi iqembu elilula likaLie lingaqukatha amaqembu amancane avamile. Ngenxa yalesi sizathu, incazelo yeqembu elula likaLie ayifani nencazelo yeqembu likaLie elilula njengeqembu elingaqondakali.  Amaqembu kaLie alula ahlanganisa amaqembu amaningi kaLie akudala, anikeza isisekelo seqembu-theoretic se-geometry eyindilinga, i-geometry ye-projective kanye nama-geometry ahlobene ngomqondo wohlelo lukaFelix Klein's Erlangen. Kwavela ngesikhathi sokuhlukaniswa kwamaqembu alula ka-Lie ukuthi kukhona namathuba ambalwa angavamile angahambisani nanoma iyiphi i-geometry ejwayelekile. La maqembu ayingqayizivele anezibonelo eziningi ezikhethekile nokwakheka kwamanye amagatsha ezibalo, kanye ne-theoretical physics yanamuhla.  Njengesibonelo esiphikisayo, iqembu lomugqa jikelele alilula, futhi alilula. Lokhu kungenxa yokuthi iziphindaphindo zobunikazi zakha i-non-trivial subgroup evamile, ngaleyo ndlela igwema incazelo. Ngokulinganayo, i- Lie algebra ehambisanayo inefomu lokubulala eliwohlokayo, ngoba iziphindaphindo ze- identity map kuya ku- zero element ye- algebra. Ngakho-ke, i- Lie algebra ehambisanayo nayo ayilula noma ilula. Esinye isibonelo esiphikisayo ngamaqembu akhethekile angama-orthogonal anobukhulu obulinganayo. Lezi zine-matrix -  -I maphakathi, futhi lesi sici sixhumene nomzila wesici sobunikazi, ngakho la maqembu ayayibalekela incazelo. Zombili lezi ziyiqembu elinciphisayo.