[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/en\/wiki19\/complex-lie-algebra-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/en\/wiki19\/complex-lie-algebra-wikipedia\/","headline":"Complex Lie algebra – Wikipedia","name":"Complex Lie algebra – Wikipedia","description":"before-content-x4 From Wikipedia, the free encyclopedia after-content-x4 In mathematics, a complex Lie algebra is a Lie algebra over the complex","datePublished":"2019-04-13","dateModified":"2019-04-13","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/en\/wiki19\/author\/lordneo\/#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/en\/wiki19\/author\/lordneo\/","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/40a913b1503ed9ec94361b99f7fd59ef60705c28","url":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/40a913b1503ed9ec94361b99f7fd59ef60705c28","height":"","width":""},"url":"https:\/\/wiki.edu.vn\/en\/wiki19\/complex-lie-algebra-wikipedia\/","about":["Wiki"],"wordCount":6535,"articleBody":"     (adsbygoogle = window.adsbygoogle || []).push({});before-content-x4From Wikipedia, the free encyclopedia       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4In mathematics, a complex Lie algebra is a Lie algebra over the complex numbers.Given a complex Lie algebra        (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4g{displaystyle {mathfrak {g}}}, its conjugate g\u00af{displaystyle {overline {mathfrak {g}}}} is a complex Lie algebra with the same underlying real vector space but with i=\u22121{displaystyle i={sqrt {-1}}} acting as \u2212i{displaystyle -i} instead.[1] As a real Lie algebra, a complex Lie algebra        (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4g{displaystyle {mathfrak {g}}} is trivially isomorphic to its conjugate. A complex Lie algebra is isomorphic to its conjugate if and only if it admits a real form (and is said to be defined over the real numbers).Real form[edit]Given a complex Lie algebra g{displaystyle {mathfrak {g}}}, a real Lie algebra g0{displaystyle {mathfrak {g}}_{0}} is said to be a real form of g{displaystyle {mathfrak {g}}} if the complexification g0\u2297RC{displaystyle {mathfrak {g}}_{0}otimes _{mathbb {R} }mathbb {C} } is isomorphic to g{displaystyle {mathfrak {g}}}.A real form g0{displaystyle {mathfrak {g}}_{0}} is abelian (resp. nilpotent, solvable, semisimple) if and only if g{displaystyle {mathfrak {g}}} is abelian (resp. nilpotent, solvable, semisimple).[2] On the other hand, a real form g0{displaystyle {mathfrak {g}}_{0}} is simple if and only if either g{displaystyle {mathfrak {g}}} is simple or g{displaystyle {mathfrak {g}}} is of the form s\u00d7s\u00af{displaystyle {mathfrak {s}}times {overline {mathfrak {s}}}} where s,s\u00af{displaystyle {mathfrak {s}},{overline {mathfrak {s}}}} are simple and are the conjugates of each other.[2]The existence of a real form in a complex Lie algebra g{displaystyle {mathfrak {g}}} implies that g{displaystyle {mathfrak {g}}} is isomorphic to its conjugate;[1] indeed, if g=g0\u2297RC=g0\u2295ig0{displaystyle {mathfrak {g}}={mathfrak {g}}_{0}otimes _{mathbb {R} }mathbb {C} ={mathfrak {g}}_{0}oplus i{mathfrak {g}}_{0}}, then let \u03c4:g\u2192g\u00af{displaystyle tau :{mathfrak {g}}to {overline {mathfrak {g}}}} denote the R{displaystyle mathbb {R} }-linear isomorphism induced by complex conjugate and then\u03c4(i(x+iy))=\u03c4(ix\u2212y)=\u2212ix\u2212y=\u2212i\u03c4(x+iy){displaystyle tau (i(x+iy))=tau (ix-y)=-ix-y=-itau (x+iy)},which is to say \u03c4{displaystyle tau } is in fact a C{displaystyle mathbb {C} }-linear isomorphism.Conversely, suppose there is a C{displaystyle mathbb {C} }-linear isomorphism \u03c4:g\u2192\u223cg\u00af{displaystyle tau :{mathfrak {g}}{overset {sim }{to }}{overline {mathfrak {g}}}}; without loss of generality, we can assume it is the identity function on the underlying real vector space. Then define g0={z\u2208g|\u03c4(z)=z}{displaystyle {mathfrak {g}}_{0}={zin {mathfrak {g}}|tau (z)=z}}, which is clearly a real Lie algebra. Each element z{displaystyle z} in g{displaystyle {mathfrak {g}}} can be written uniquely as z=2\u22121(z+\u03c4(z))+i2\u22121(i\u03c4(z)\u2212iz){displaystyle z=2^{-1}(z+tau (z))+i2^{-1}(itau (z)-iz)}. Here, \u03c4(i\u03c4(z)\u2212iz)=\u2212iz+i\u03c4(z){displaystyle tau (itau (z)-iz)=-iz+itau (z)} and similarly \u03c4{displaystyle tau } fixes z+\u03c4(z){displaystyle z+tau (z)}. Hence, g=g0\u2295ig0{displaystyle {mathfrak {g}}={mathfrak {g}}_{0}oplus i{mathfrak {g}}_{0}}; i.e., g0{displaystyle {mathfrak {g}}_{0}} is a real form.Complex Lie algebra of a complex Lie group[edit]Let g{displaystyle {mathfrak {g}}} be a semisimple complex Lie algebra that is the Lie algebra of a complex Lie group G{displaystyle G}. Let h{displaystyle {mathfrak {h}}} be a Cartan subalgebra of g{displaystyle {mathfrak {g}}} and H{displaystyle H} the Lie subgroup corresponding to h{displaystyle {mathfrak {h}}}; the conjugates of H{displaystyle H} are called Cartan subgroups.Suppose there is the decomposition g=n\u2212\u2295h\u2295n+{displaystyle {mathfrak {g}}={mathfrak {n}}^{-}oplus {mathfrak {h}}oplus {mathfrak {n}}^{+}} given by a choice of positive roots. Then the exponential map defines an isomorphism from n+{displaystyle {mathfrak {n}}^{+}} to a closed subgroup U\u2282G{displaystyle Usubset G}.[3] The Lie subgroup B\u2282G{displaystyle Bsubset G} corresponding to the Borel subalgebra b=h\u2295n+{displaystyle {mathfrak {b}}={mathfrak {h}}oplus {mathfrak {n}}^{+}} is closed and is the semidirect product of H{displaystyle H} and U{displaystyle U};[4] the conjugates of B{displaystyle B} are called Borel subgroups.^ a b Knapp 2002, Ch. VI, \u00a7 9.^ a b Serre 2001, Ch. II, \u00a7 8, Theorem 9.^ Serre 2001, Ch. VIII, \u00a7 4, Theorem 6 (a).^ Serre 2001, Ch. VIII, \u00a7 4, Theorem 6 (b).References[edit]       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4"},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki19\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki19\/complex-lie-algebra-wikipedia\/#breadcrumbitem","name":"Complex Lie algebra – Wikipedia"}}]}]