Real form (Lie theory)

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the notion of a real form relates objects defined over the <a href="/facts/Field_(algebra)/xAjAS4ko">field</a> of <a href="/facts/Real_number/R02gw5Pb">real</a> and <a href="/facts/Complex_number/2w7mqI7e">complex</a> numbers. A real <a href="/facts/Lie_algebra/n2gAUkNq">Lie algebra</a> g0 is called a real form of a <a href="/facts/Complex_Lie_algebra/qCcze5of">complex Lie algebra</a> g if g is the <a href="/facts/Complexification/fDfFLccF">complexification</a> of g0:

g
          
        
        ≃
        
          
            
              g
            
          
          
            0
          
        
        
          ⊗
          
            
              R
            
          
        
        
          C
        
        .
      
    
    {\displaystyle {\mathfrak {g}}\simeq {\mathfrak {g}}_{0}\otimes _{\mathbb {R} }\mathbb {C} .}

The notion of a real form can also be defined for complex <a href="/facts/Lie_group/a9jCQyjF">Lie groups</a>. Real forms of complex <a href="/facts/Semisimple_Lie_group/m8jA3Bcf">semisimple Lie groups</a> and Lie algebras have been completely classified by <a href="/facts/%25C3%2589lie_Cartan/Uvhkp66S">Élie Cartan</a>.

Real form (Lie theory) open-in-new

Real form (Lie theory)