mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, the notion of a real form relates objects defined over the field of real and

complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...

numbers. A real

Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...

''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 ins ...

''g'' if ''g'' is the

complexification In mathematics, the complexification of a vector space over the field of real numbers (a "real vector space") yields a vector space over the complex number field, obtained by formally extending the scaling of vectors by real numbers to include ...

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 (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Eucli ...

s. Real forms of complex

semisimple Lie group In mathematics, a simple Lie group is a connected space, connected nonabelian group, non-abelian Lie group ''G'' which does not have nontrivial connected normal subgroups. The list of simple Lie groups can be used to read off the list of simple ...

s 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. He ...

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 ...

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 which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...

Classification

Just as complex

semisimple Lie algebra In mathematics, a Lie algebra is semisimple if it is a direct sum of modules, direct sum of Simple Lie algebra, simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper Lie algebra#Subalgebras.2C ideals ...

s are classified by

Dynkin diagram In the Mathematics, mathematical field of Lie theory, a Dynkin diagram, named for Eugene Dynkin, is a type of Graph (discrete mathematics), graph with some edges doubled or tripled (drawn as a double or triple line). Dynkin diagrams arise in the ...

s, 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

s 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 In mathematics, the special linear group \operatorname(n,R) of degree n over a commutative ring R is the set of n\times n Matrix (mathematics), matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix ...

''SL''(''n'',C), the compact real form is the

special unitary group In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1. The matrices of the more general unitary group may have complex determinants with absolute value 1, rather than real 1 ...

''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

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 isometries contains an inversion symmetry about every point. This can be studied with the tools of Riemannian geomet ...

s. In general, there may be more than two real forms. Suppose that ''g''₀ is a

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 Sylvester's law of inertia is a theorem in matrix algebra about certain properties of the coefficient matrix of a real quadratic form that remain invariant under a change of basis. Namely, if A is a symmetric matrix, then for any invertible matr ...

, 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 Split(s) or The Split may refer to: Places * Split, Croatia, the largest coastal city in Croatia * Split Island, Canada, an island in the Hudson Bay * Split Island, Falkland Islands * Split Island, Fiji, better known as Hạfliua Arts, enter ...

, or normal, if in each

Cartan decomposition In mathematics, the Cartan decomposition is a decomposition of a semisimple Lie group or Lie algebra, which plays an important role in their structure theory and representation theory. It generalizes the polar decomposition or singular value deco ...

''g''₀ = ''k''₀ ⊕ ''p''₀, the space ''p''₀ contains a maximal abelian subalgebra of ''g''₀, i.e. its

Cartan subalgebra In mathematics, a Cartan subalgebra, often abbreviated as CSA, is a nilpotent subalgebra \mathfrak of a Lie algebra \mathfrak that is self-normalising (if ,Y\in \mathfrak for all X \in \mathfrak, then Y \in \mathfrak). They were introduced by ...

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 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, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...

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) sho ...

is negative definite, i.e. the index of ''g''₀ is zero. In this case ''g''₀ = ''k''₀ is a compact Lie algebra. It is known that under the Lie correspondence, compact Lie algebras correspond to

compact Lie group In mathematics, a compact (topological) group is a topological group whose topology realizes it as a compact space, compact topological space (when an element of the group is operated on, the result is also within the group). Compact groups are ...

s. 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 algebra The classical Lie algebras are finite-dimensional Lie algebras over a field which can be classified into four types A_n , B_n , C_n and D_n , where for \mathfrak(n) the general linear Lie algebra and I_n the n \times n identity matrix: ...

s 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

: :

\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 (making it a compact Lie algebra), and that the complexification of ''u''₀ is ''g''. Therefore, ''u''₀ is a compact form of ''g''.

Notes

References

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