mathematical
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 ...
field of
Lie theory
In mathematics, the mathematician Sophus Lie ( ) initiated lines of study involving integration of differential equations, transformation groups, and contact of spheres that have come to be called Lie theory. For instance, the latter subject is ...
, 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
In mathematics, a compact (topological) group is a topological group whose topology realizes it as a compact topological space (when an element of the group is operated on, the result is also within the group). Compact groups are a natural gen ...
; this definition includes tori. Intrinsically and algebraically, a compact Lie algebra is a real Lie algebra whose 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 ...
; this definition is more restrictive and excludes tori,. A compact Lie algebra can be seen as the smallest
real form
In mathematics, the notion of a real form relates objects defined over the field of real and complex numbers. A real Lie algebra ''g''0 is called a real form of a complex Lie algebra ''g'' if ''g'' is the complexification of ''g''0:
: \mathf ...
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 ''semi''definite, 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 ''semisimple'' Lie group.
In general, the Lie algebra of a compact Lie group decomposes as the Lie algebra direct sum of a commutative summand (for which the corresponding subgroup is a torus) and a summand on which the Killing form is negative definite.
It is important to note that the converse of the first result above is false: Even if the Killing form of a Lie algebra is negative semidefinite, this does not mean that the Lie algebra is the Lie algebra of some compact group. For example, the Killing form on the Lie algebra of the Heisenberg group is identically zero, hence negative semidefinite, but this Lie algebra is not the Lie algebra of any compact group.
Properties
* Compact Lie algebras are reductive; note that the analogous result is true for compact groups in general.
*The Lie algebra for the compact Lie group ''G'' admits an Ad(''G'')-invariant
inner product
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
,. Conversely, if admits an Ad-invariant inner product, then is the Lie algebra of some compact group. If is semisimple, this inner product can be taken to be the negative of the Killing form. Thus relative to this inner product, Ad(''G'') acts by
orthogonal transformation In linear algebra, an orthogonal transformation is a linear transformation ''T'' : ''V'' → ''V'' on a real inner product space ''V'', that preserves the inner product. That is, for each pair of elements of ''V'', we h ...
s () and acts by
skew-symmetric matrices
In mathematics, particularly in linear algebra, a skew-symmetric (or antisymmetric or antimetric) matrix is a square matrix whose transpose equals its negative. That is, it satisfies the condition
In terms of the entries of the matrix, if a_ ...
(). It is possible to develop the theory of complex semisimple Lie algebras by viewing them as the complexifications of Lie algebras of compact groups; Chapter 7 the existence of an Ad-invariant inner product on the compact real form greatly simplifies the development.
*:This can be seen as a compact analog of
Ado's theorem In abstract algebra, Ado's theorem is a theorem characterizing finite-dimensional Lie algebras.
Statement
Ado's theorem states that every finite-dimensional Lie algebra ''L'' over a field ''K'' of characteristic zero can be viewed as a Lie algebr ...
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 In the mathematics, mathematical study of Lie algebras and Lie groups, a Satake diagram is a generalization of a Dynkin diagram introduced by whose configurations classify semisimple Lie algebra, simple Lie algebras over the field (mathematics), fi ...
of a compact Lie algebra is the
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 ...
of the complex Lie algebra with ''all'' vertices blackened.
* Compact Lie algebras are opposite to split real Lie algebras among
real form
In mathematics, the notion of a real form relates objects defined over the field of real and complex numbers. A real Lie algebra ''g''0 is called a real form of a complex Lie algebra ''g'' if ''g'' is the complexification of ''g''0:
: \mathf ...
s, 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 form
In mathematics, the notion of a real form relates objects defined over the Field (algebra), field of Real number, real and Complex number, complex numbers. A real Lie algebra ''g''0 is called a real form of a complex Lie algebra ''g'' if ''g'' is ...
s of the complex
semisimple Lie algebra
In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper ideals).
Throughout the article, unless otherwise stated, a Lie algebra is ...
s. These are:
* corresponding to the
special unitary group
In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1.
The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the special ...
(properly, the compact form is PSU, the
projective special unitary group In mathematics, the projective unitary group is the quotient of the unitary group by the right multiplication of its center, , embedded as scalars.
Abstractly, it is the holomorphic isometry group of complex projective space, just as the projectiv ...
compact symplectic group
In mathematics, the name symplectic group can refer to two different, but closely related, collections of mathematical groups, denoted and for positive integer ''n'' and field F (usually C or R). The latter is called the compact symplectic gro ...
projective special orthogonal group In projective geometry and linear algebra, the projective orthogonal group PO is the induced action of the orthogonal group of a quadratic space ''V'' = (''V'',''Q'')A quadratic space is a vector space ''V'' together with a quadratic form ''Q''; th ...
);
* Compact real forms of the exceptional Lie algebras
Isomorphisms
The classification is non-redundant if one takes for for for and for If one instead takes or one obtains certain
exceptional isomorphism In mathematics, an exceptional isomorphism, also called an accidental isomorphism, is an isomorphism between members ''a'i'' and ''b'j'' of two families, usually infinite, of mathematical objects, that is not an example of a pattern of such is ...
s.
For is the trivial diagram, corresponding to the trivial group
For the isomorphism corresponds to the isomorphisms of diagrams and the corresponding isomorphisms of Lie groups (the 3-sphere or
unit quaternion
In mathematics, a versor is a quaternion of norm one (a ''unit quaternion''). The word is derived from Latin ''versare'' = "to turn" with the suffix ''-or'' forming a noun from the verb (i.e. ''versor'' = "the turner"). It was introduced by Will ...
s).
For the isomorphism corresponds to the isomorphisms of diagrams and the corresponding isomorphism of Lie groups
For the isomorphism corresponds to the isomorphisms of diagrams and the corresponding isomorphism of Lie groups
If one considers and as diagrams, these are isomorphic to and respectively, with corresponding isomorphisms of Lie algebras.
See also
*
Real form
In mathematics, the notion of a real form relates objects defined over the field of real and complex numbers. A real Lie algebra ''g''0 is called a real form of a complex Lie algebra ''g'' if ''g'' is the complexification of ''g''0:
: \mathf ...
*
Split Lie algebra
In the mathematical field of Lie theory, a split Lie algebra is a pair (\mathfrak, \mathfrak) where \mathfrak is a Lie algebra and \mathfrak < \mathfrak is a splitting