split real form
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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the notion of a real form relates objects defined over the
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
of real and complex numbers. A real
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
''g''0 is called a real form of a
complex Lie algebra In mathematics, a complex Lie algebra is a Lie algebra over the complex numbers. Given a complex Lie algebra \mathfrak, its conjugate \overline is a complex Lie algebra with the same underlying real vector space but with i = \sqrt acting as -i inst ...
''g'' if ''g'' is the complexification of ''g''0: : \mathfrak\simeq\mathfrak_0\otimes_\mathbb. The notion of a real form can also be defined for complex
Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
s. Real forms of complex semisimple Lie groups and Lie algebras have been completely classified by
Élie Cartan Élie Joseph Cartan (; 9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups, differential systems (coordinate-free geometric formulation of PDEs), and differential geometry. ...
.


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 group In mathematics, a linear algebraic group is a subgroup of the group of invertible n\times n matrices (under matrix multiplication) that is defined by polynomial equations. An example is the orthogonal group, defined by the relation M^TM = I_n wh ...
s, the notions of complexification and real form have a natural description in the language of
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
.


Classification

Just as complex semisimple Lie algebras are classified by Dynkin diagrams, the real forms of a semisimple Lie algebra are classified by
Satake diagram In the mathematical study of Lie algebras and Lie groups, a Satake diagram is a generalization of a Dynkin diagram introduced by whose configurations classify simple Lie algebras over the field of real numbers. The Satake diagrams associated to a D ...
s, 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 Élie Joseph Cartan (; 9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups, differential systems (coordinate-free geometric formulation of PDEs), and differential geometry. ...
in the context of
Riemannian symmetric space In mathematics, a symmetric space is a Riemannian manifold (or more generally, a pseudo-Riemannian manifold) whose group of symmetries contains an inversion symmetry about every point. This can be studied with the tools of Riemannian geometry, ...
s. 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 Élie Joseph Cartan (; 9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups, differential systems (coordinate-free geometric formulation of PDEs), and differential geometry. ...
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 In the mathematical study of Lie algebras and Lie groups, a Satake diagram is a generalization of a Dynkin diagram introduced by whose configurations classify simple Lie algebras over the field of real numbers. The Satake diagrams associated to a D ...
with no vertices blackened and no arrows.


Compact real form

A real Lie algebra ''g''0 is called
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...
if the
Killing form In mathematics, the Killing form, named after Wilhelm Killing, is a symmetric bilinear form that plays a basic role in the theories of Lie groups and Lie algebras. Cartan's criteria (criterion of solvability and criterion of semisimplicity) show ...
is
negative definite In mathematics, negative definiteness is a property of any object to which a bilinear form may be naturally associated, which is negative-definite. See, in particular: * Negative-definite bilinear form * Negative-definite quadratic form * Nega ...
, i.e. the index of ''g''0 is zero. In this case ''g''0 = ''k''0 is a
compact Lie algebra In the mathematical field of Lie theory, there are two definitions of a compact Lie algebra. Extrinsically and topologically, a compact Lie algebra is the Lie algebra of a compact Lie group; this definition includes tori. Intrinsically and algeb ...
. It is known that under the
Lie correspondence A lie is an assertion that is believed to be false, typically used with the purpose of deceiving or misleading someone. The practice of communicating lies is called lying. A person who communicates a lie may be termed a liar. Lies can be inter ...
, compact Lie algebras correspond to compact Lie groups. The compact form corresponds to the
Satake diagram In the mathematical study of Lie algebras and Lie groups, a Satake diagram is a generalization of a Dynkin diagram introduced by whose configurations classify simple Lie algebras over the field of real numbers. The Satake diagrams associated to a D ...
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 ^. 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_0=\mathfrak_0\oplus\mathfrak_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_0=\mathfrak_0\oplus i\mathfrak_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 In mathematics, negative definiteness is a property of any object to which a bilinear form may be naturally associated, which is negative-definite. See, in particular: * Negative-definite bilinear form * Negative-definite quadratic form * Nega ...
(making it a compact Lie algebra), and that the complexification of ''u''0 is ''g''. Therefore, ''u''0 is a compact form of ''g''.


See also

*
Complexification (Lie group) In mathematics, the complexification or universal complexification of a real Lie group is given by a continuous homomorphism of the group into a complex Lie group with the universal property that every continuous homomorphism of the original grou ...


Notes


References

* * {{DEFAULTSORT:Real Form (Lie Theory) Lie groups Lie algebras