title: Real form (Lie theory)
text: # Real form (Lie theory)
  
  Lie groups and Lie algebras :
  
  Classical groups
  
  General linear GL(n)
  
  Special linear SL(n)
  
  Orthogonal O(n)
  
  Special orthogonal SO(n)
  
  Unitary U(n)
  
  Special unitary SU(n)
  
  Symplectic Sp(n) :
  
  Simple Lie groups
  
  Classical
  
  $A_n$
  
  $B_n$
  
  $C_n$
  
  $D_n$
  
  Exceptional
  
  $G_2$
  
  $F_4$
  
  $E_6$
  
  $E_7$
  
  $E_8$ :
  
  Other Lie groups
  
  Circle
  
  Lorentz
  
  Poincaré
  
  Conformal group
  
  Diffeomorphism
  
  Loop
  
  Euclidean :
  
  Lie algebras
  
  Lie group–Lie algebra correspondence
  
  Exponential map
  
  Adjoint representation
  
  Killing formIndex
  
  Simple Lie algebra
  
  Loop algebra
  
  Affine Lie algebra :
  
  Semisimple Lie algebra
  
  Dynkin diagrams
  
  Cartan subalgebra
  
  Root systemWeyl group
  
  Real formComplexification
  
  Split Lie algebra
  
  Compact Lie algebra :
  
  Representation theory
  
  Lie group representation
  
  Lie algebra representation
  
  Representation theory of semisimple Lie algebras
  
  Representations of classical Lie groups
  
  Theorem of the highest weight
  
  Borel–Weil–Bott theorem :
  
  Lie groups in physics
  
  Particle physics and representation theory
  
  Lorentz group representations
  
  Poincaré group representations
  
  Galilean group representations :
  
  Scientists
  
  Sophus Lie
  
  Henri Poincaré
  
  Wilhelm Killing
  
  Élie Cartan
  
  Hermann Weyl
  
  Claude Chevalley
  
  Harish-Chandra
  
  Armand Borel :
  
  Glossary
  
  Table of Lie groups :
  
  vte :
  
  In mathematics, the notion of a real form relates objects defined over the field of real and complex numbers. A real Lie algebra $g_0$ is called a real form of a complex Lie algebra g if g is the complexification of $g_0:$
  
      ${\mathfrak {g}}\simeq {\mathfrak {g}}_{0}\otimes _{\mathbb {R} }\mathbb {C} .$
  
  The notion of a real form can also be defined for complex Lie groups. Real forms of complex semisimple Lie groups and Lie algebras have been completely classified by Élie Cartan.
  
  ## Real forms for Lie groups and algebraic groups
  
  Using the Lie correspondence between Lie groups and Lie algebras, the notion of a real form can be defined for Lie groups. In the case of linear algebraic groups, the notions of complexification and real form have a natural description in the language of algebraic geometry.
  
  ## Classification
  
  Just as complex semisimple Lie algebras are classified by Dynkin diagrams, the real forms of a semisimple Lie algebra are classified by Satake diagrams, which are obtained from the Dynkin diagram of the complex form by labeling some vertices black (filled), and connecting some other vertices in pairs by arrows, according to certain rules.
  
  It is a basic fact in the structure theory of complex semisimple Lie algebras that every such algebra has two special real forms: one is the compact real form and corresponds to a compact Lie group under the Lie correspondence (its Satake diagram has all vertices blackened), and the other is the split real form and corresponds to a Lie group that is as far as possible from being compact (its Satake diagram has no vertices blackened and no arrows). In the case of the complex special linear group SL(n,C), the compact real form is the special unitary group SU(n) and the split real form is the real special linear group SL(n,R). The classification of real forms of semisimple Lie algebras was accomplished by Élie Cartan in the context of Riemannian symmetric spaces. In general, there may be more than two real forms.
  
  Suppose that $g_0$ is a semisimple Lie algebra over the field of real numbers. By Cartan's criterion, the Killing form is nondegenerate, and can be diagonalized in a suitable basis with the diagonal entries +1 or −1. By Sylvester's law of inertia, the number of positive entries, or the positive index of inertia, is an invariant of the bilinear form, i.e. it does not depend on the choice of the diagonalizing basis. This is a number between 0 and the dimension of g which is an important invariant of the real Lie algebra, called its index.
  
  ### Split real form
  
  A real form $g_0$ of a finite-dimensional complex semisimple Lie algebra g is said to be split, or normal, if in each Cartan decomposition $g_0$ = $k_0$ ⊕ $p_0,$ the space $p_0$ contains a maximal abelian subalgebra of $g_0,$ i.e. its Cartan subalgebra. Élie Cartan proved that every complex semisimple Lie algebra g has a split real form, which is unique up to isomorphism. It has maximal index among all real forms.
  
  The split form corresponds to the Satake diagram with no vertices blackened and no arrows.
  
  ### Compact real form
  
  A real Lie algebra $g_0$ is called compact if the Killing form is negative definite, i.e. the index of $g_0$ is zero. In this case $g_0$ = $k_0$ is a compact Lie algebra. It is known that under the Lie correspondence, compact Lie algebras correspond to compact Lie groups.
  
  The compact form corresponds to the Satake diagram with all vertices blackened.
  
  ## Construction of the compact real form
  
  In general, the construction of the compact real form uses structure theory of semisimple Lie algebras. For classical Lie algebras there is a more explicit construction.
  
  Let $g_0$ be a real Lie algebra of matrices over R that is closed under the transpose map,
  
      $X\to {X}^{t}.$
  
  Then $g_0$ decomposes into the direct sum of its skew-symmetric part $k_0$ and its symmetric part $p_0.$ This is the Cartan decomposition:
  
      ${\mathfrak {g}}_{0}={\mathfrak {k}}_{0}\oplus {\mathfrak {p}}_{0}.$
  
  The complexification g of $g_0$ decomposes into the direct sum of $g_0$ and $ig_0.$ The real vector space of matrices
  
      ${\mathfrak {u}}_{0}={\mathfrak {k}}_{0}\oplus i{\mathfrak {p}}_{0}$
  
  is a subspace of the complex Lie algebra g that is closed under the commutators and consists of skew-hermitian matrices. It follows that $u_0$ is a real Lie subalgebra of g, that its Killing form is negative definite (making it a compact Lie algebra), and that the complexification of $u_0$ is g. Therefore, $u_0$ is a compact form of g.
  
  