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 An Bn Cn Dn Exceptional G2 F4 E6 E7 E8
Other Lie groups Circle Lorentz Poincaré Conformal group Diffeomorphism Loop Euclidean
Lie algebras Lie group–Lie algebra correspondence Exponential map Adjoint representation Killing form Index Simple Lie algebra Loop algebra Affine Lie algebra
Semisimple Lie algebra Dynkin diagrams Cartan subalgebra Root system Weyl group Real form Complexification 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
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In mathematics, the notion of a real form relates objects defined over the field of real and complex numbers. A real Lie algebra g₀ is called a real form of a complex Lie algebra g if g is the complexification of g₀:

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

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.

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.

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 g₀ 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.

A real form g₀ of a finite-dimensional complex semisimple Lie algebra g is said to be split, or normal, if in each Cartan decomposition g₀ = k₀ ⊕ p₀, the space p₀ contains a maximal abelian subalgebra of g₀, 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.^[1] It has maximal index among all real forms.

The split form corresponds to the Satake diagram with no vertices blackened and no arrows.

A real Lie algebra g₀ is called compact if the Killing form is negative definite, i.e. the index of g₀ is zero. In this case g₀ = k₀ 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.

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 g₀ be a real Lie algebra of matrices over R that is closed under the transpose map,

${\displaystyle X\to {X}^{t}.}$

Then g₀ decomposes into the direct sum of its skew-symmetric part k₀ and its symmetric part p₀. This is the Cartan decomposition:

${\displaystyle {\mathfrak {g}}_{0}={\mathfrak {k}}_{0}\oplus {\mathfrak {p}}_{0}.}$

The complexification g of g₀ decomposes into the direct sum of g₀ and ig₀. The real vector space of matrices

${\displaystyle {\mathfrak {u}}_{0}={\mathfrak {k}}_{0}\oplus i{\mathfrak {p}}_{0}}$

is a subspace of the complex Lie algebra g that is closed under the commutators and consists of skew-hermitian matrices. It follows that u₀ 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 u₀ is g. Therefore, u₀ is a compact form of g.

• Complexification (Lie group)