Real form

The term real form is used in mathematics to relate objects , in particular algebraic structures , defined using real and complex numbers . It is mainly used in the theory of Lie algebras and Lie groups .

A real Lie algebra is a real form of a complex Lie algebra if the complexification of is, that is ${\ displaystyle {\ mathfrak {g}}}$ ${\ displaystyle {\ mathfrak {g}} _ {\ mathbb {C}}}$ ${\ displaystyle {\ mathfrak {g}} _ {\ mathbb {C}}}$ ${\ displaystyle {\ mathfrak {g}}}$

{\ displaystyle {\ mathfrak {g}} _ {\ mathbb {C}} = {\ mathfrak {g}} \ otimes _ {\ mathbb {R}} \ mathbb {C}}

.

In general, a real form of a complex vector space can be defined by the condition . A complex vector space has an infinite number of real forms, for example or are real forms of . ${\ displaystyle V}$ ${\ displaystyle W}$ ${\ displaystyle V \ otimes _ {\ mathbb {R}} \ mathbb {C} = W}$ ${\ displaystyle \ mathbb {R} ^ {n}}$ ${\ displaystyle (i \ mathbb {R}) ^ {n}}$ ${\ displaystyle \ mathbb {C} ^ {n}}$

A real form of a complex Lie group is a subgroup whose Lie algebra is a real form of the Lie algebra of the complex Lie group.

Semi-simple Lie algebras

Every semi-simple complex Lie algebra has at least two real forms . ${\ displaystyle {\ mathfrak {g}} _ {\ mathbb {C}}}$ ${\ displaystyle {\ mathfrak {g}}}$

One of the two real forms is a compact Lie algebra; H. the killing form is negative definite .

The other of the two real forms is a fissile Lie algebra; H. there is a Cartan subalgebra such that the adjoint mapping is diagonalizable for all of them . ${\ displaystyle {\ mathfrak {h}} \ subset {\ mathfrak {g}}}$ ${\ displaystyle h \ in {\ mathfrak {h}}}$ ${\ displaystyle ad (h) \ colon {\ mathfrak {g}} \ to {\ mathfrak {g}}}$

In general it can have other real forms. ${\ displaystyle {\ mathfrak {g}} _ {\ mathbb {C}}}$

Examples

For a semi-simple complex Lie algebra , the following list first names the compact, then the fissile real form. ${\ displaystyle {\ mathfrak {g}} _ {\ mathbb {C}}}$

{\ displaystyle sl (n, \ mathbb {C}): su (n), sl (n, \ mathbb {R})}

{\ displaystyle so (2n + 1, \ mathbb {C}): so (2n + 1), so (n, n + 1, \ mathbb {C})}

{\ displaystyle so (2n, \ mathbb {C}): so (2n), so (n, n, \ mathbb {C})}

${\ displaystyle sp (n, \ mathbb {C}): sp (n), sp (n, \ mathbb {R})}$ (where the Lie algebra denotes the compact symplectic group ) ${\ displaystyle sp (n)}$

classification

Real forms of a semi-simple complex Lie algebra are classified by Satake diagrams , certain refinements of the Dynkin diagram by . ${\ displaystyle {\ mathfrak {g}} _ {\ mathbb {C}}}$ ${\ displaystyle {\ mathfrak {g}} _ {\ mathbb {C}}}$

Representation theory

The complex representations of correspond 1: 1 to the complex representations of : one obtains all representations of by restricting the representations of the complexized Lie algebra. For example, the representation theory of is equivalent to the representation theory of sl (2, C) . ${\ displaystyle {\ mathfrak {g}}}$ ${\ displaystyle {\ mathfrak {g}} \ otimes \ mathbb {C}}$ ${\ displaystyle {\ mathfrak {g}}}$ ${\ displaystyle su (2)}$

literature

Bourbaki, Nicolas: VIII: Split Semi-simple Lie Algebras , Elements of Mathematics: Lie Groups and Lie Algebras. (Chapters 7-9)
Onishchik, AL; Vinberg, Ėrnest Borisovich: Lie groups and Lie algebras III: structure of Lie groups and Lie algebras (Chapter 4.4: "Split Real Semisimple Lie Algebras")