Compact Lie algebra - AbsoluteAstronomy.com

In the mathematical field of Lie theory

Lie theory

Lie theory is an area of mathematics, developed initially by Sophus Lie.Early expressions of Lie theory are found in books composed by Lie with Friedrich Engel and Georg Scheffers from 1888 to 1896....

, a Lie algebra

Lie algebra

In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...

is compact if it is the Lie algebra of a compact Lie group. Intrinsically, a compact Lie algebra is a real Lie algebra whose Killing form

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...

is negative definite, though this definition does not quite agree with the previous. A compact Lie algebra can be seen as the smallest real form of a corresponding complex Lie algebra, namely the complexification.

Definition

Formally, one may define a compact Lie algebra either as the Lie algebra of a compact Lie group, or as a real Lie algebra whose Killing form is negative definite. These definitions do not quite agree:

The Killing form on the Lie algebra of a compact Lie group is negative semidefinite, not negative definite in general.
If the Killing form of a Lie algebra is negative definite, then the Lie algebra is the Lie algebra of a compact Lie group.

The difference is precisely in whether to include tori (and their corresponding Lie algebra, which is abelian and hence has trivial Killing form) or not: real Lie algebras with negative definite Killing forms correspond to compact semisimple Lie groups, while real Lie algebras with negative semidefinite Killing forms correspond to products of compact semisimple Lie groups and tori. One can distinguish between these by calling a Lie algebra with negative semidefinite Killing form a compact reductive Lie algebra, and a Lie algebra with negative definite Killing form a compact semisimple Lie algebra, which corresponds to reductive Lie algebras being direct sums of semisimple and abelian.

Properties

Compact Lie algebras are reductive; note that the analogous result is true for compact groups in general.
A compact Lie algebra for the compact Lie group G admits an Ad(G)-invariant inner product, and this property characterizes compact Lie algebras. This inner product can be taken to be the negative of the Killing form, and this is the unique Ad(G)-invariant inner product up to scale. Thus relative to this inner product, Ad(G) acts by orthogonal transformations () and acts by skew-symmetric matrices ().

This can be seen as a compact analog of Ado's theorem
Ado's theorem
In abstract algebra, Ado's theorem states that every finite-dimensional Lie algebra L over a field K of characteristic zero can be viewed as a Lie algebra of square matrices under the commutator bracket...

on the representability of Lie algebras: just as every finite-dimensional Lie algebra in characteristic 0 embeds in every compact Lie algebra embeds in
The Satake diagram
Satake diagram
In the mathematical study of Lie algebras and Lie groups, a Satake diagram is a generalization of a Dynkin diagram whose configurations classify simple Lie algebras over the field of real numbers...

of a compact Lie algebra is the Dynkin diagram of the complex Lie algebra with all vertices blackened.
Compact Lie algebras are opposite to split real Lie algebras among real forms, split Lie algebras being "as far as possible" from being compact.

Classification

The compact Lie algebras are classified and named according to the compact real forms of the complex semisimple Lie algebras. These are:

corresponding to the special unitary group
Special unitary group
The special unitary group of degree n, denoted SU, is the group of n×n unitary matrices with determinant 1. The group operation is that of matrix multiplication...

(properly, the compact form is PSU, the projective special unitary group);
corresponding to the special orthogonal group (or corresponding to the orthogonal group
Orthogonal group
In mathematics, the orthogonal group of degree n over a field F is the group of n × n orthogonal matrices with entries from F, with the group operation of matrix multiplication...

);
corresponding to the compact symplectic group; sometimes written ;
corresponding to the special orthogonal group (or corresponding to the orthogonal group
Orthogonal group
In mathematics, the orthogonal group of degree n over a field F is the group of n × n orthogonal matrices with entries from F, with the group operation of matrix multiplication...

) (properly, the compact form is PSO, the projective special orthogonal group);
Compact real forms of the exceptional Lie algebras

Isomorphisms

The classification is non-redundant if one takes

for

and

for

If one instead takes

one obtains certain exceptional isomorphism

Exceptional isomorphism

In mathematics, an exceptional isomorphism, also called an accidental isomorphism, is an isomorphism between members ai and bj of two families of mathematical objects, that is not an example of a pattern of such isomorphisms.Because these series of objects are presented differently, they are not...

s.

For