Real form (Lie theory)

Inmathematics,the notion of areal formrelates objects defined over thefieldofrealandcomplexnumbers. A realLie algebrag₀is called a real form of acomplex Lie algebragifgis thecomplexificationofg₀:

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

The notion of a real form can also be defined for complexLie groups.Real forms of complexsemisimple Lie groupsand Lie algebras have been completely classified byÉlie Cartan.

Real forms for Lie groups and algebraic groups

Using theLie correspondence between Lie groups and Lie algebras,the notion of a real form can be defined for Lie groups. In the case oflinear algebraic groups,the notions of complexification and real form have a natural description in the language ofalgebraic geometry.

Classification

Main article:List of simple Lie groups

Just as complexsemisimple Lie algebrasare classified byDynkin diagrams,the real forms of a semisimple Lie algebra are classified bySatake 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 complexsemisimple Lie algebrasthat every such algebra has two special real forms: one is thecompact real formand corresponds to a compact Lie group under the Lie correspondence (its Satake diagram has all vertices blackened), and the other is thesplit real formand 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 complexspecial linear groupSL(n,C), the compact real form is thespecial unitary groupSU(n) and the split real form is the real special linear groupSL(n,R). The classification of real forms of semisimple Lie algebras was accomplished byÉlie Cartanin the context ofRiemannian symmetric spaces.In general, there may be more than two real forms.

Suppose thatg₀is asemisimple Lie algebraover the field of real numbers. ByCartan's criterion,the Killing form is nondegenerate, and can be diagonalized in a suitable basis with the diagonal entries +1 or −1. BySylvester'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 ofgwhich is an important invariant of the real Lie algebra, called itsindex.

Split real form

See also:Split Lie algebra

A real formg₀of a finite-dimensional complex semisimple Lie algebragis said to besplit,ornormal,if in eachCartan decompositiong₀=k₀⊕p₀,the spacep₀contains a maximal abelian subalgebra ofg₀,i.e. itsCartan subalgebra.Élie Cartanproved that every complex semisimple Lie algebraghas a split real form, which is unique up to isomorphism.It has maximal index among all real forms.

The split form corresponds to theSatake diagramwith no vertices blackened and no arrows.

Compact real form

See also:Compact Lie algebra

A real Lie algebrag₀is calledcompactif theKilling formisnegative definite,i.e. the index ofg₀is zero. In this caseg₀=k₀is acompact Lie algebra.It is known that under theLie correspondence,compact Lie algebras correspond tocompact Lie groups.

The compact form corresponds to theSatake diagramwith 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. Forclassical Lie algebrasthere is a more explicit construction.

Letg₀be a real Lie algebra of matrices overRthat is closed under the transpose map,

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

Theng₀decomposes into the direct sum of itsskew-symmetric partk₀and itssymmetric partp₀.This is theCartan decomposition:

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

The complexificationgofg₀decomposes into the direct sum ofg₀andig₀.The real vector space of matrices

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

is a subspace of the complex Lie algebragthat is closed under the commutators and consists ofskew-hermitian matrices.It follows thatu₀is a real Lie subalgebra ofg,that its Killing form isnegative definite(making it a compact Lie algebra), and that the complexification ofu₀isg.Therefore,u₀is a compact form ofg.

See also

• Complexification (Lie group)

References

• Helgason, Sigurdur (1978),Differential geometry, Lie groups and symmetric spaces,Academic Press,ISBN0-12-338460-5
• Knapp, Anthony (2004),Lie Groups: Beyond an Introduction,Progress in Mathematics, vol. 140, Birkhäuser,ISBN0-8176-4259-5