Real form (Lie theory)

In

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:

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

The notion of a real form can also be defined for 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

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.

Split real form

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.

Compact real form

A real Lie algebra g₀ is called

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

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

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

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

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

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

is a subspace of the complex Lie algebra g that is closed under the commutators and consists of
negative definite
(making it a compact Lie algebra), and that the complexification of u₀ is g. Therefore, u₀ is a compact form of g.

See also

Complexification (Lie group)

Notes

^ Helgason 1978, p. 426

References

Helgason, Sigurdur (1978), Differential geometry, Lie groups and symmetric spaces, Academic Press,
ISBN 0-12-338460-5

Knapp, Anthony (2004), Lie Groups: Beyond an Introduction, Progress in Mathematics, vol. 140, Birkhäuser,
ISBN 0-8176-4259-5

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[1] Helgason 1978, p. 426

[1]