Lie groups andLie algebras
Classical groups General linear GL(n) Special linear SL(n) Orthogonal O(n) Special orthogonal SO(n) Unitary U(n) Special unitary SU(n) Symplectic Sp(n)
Simple Lie groups Classical An Bn Cn Dn Exceptional G2 F4 E6 E7 E8
Other Lie groups Circle Lorentz Poincaré Conformal group Diffeomorphism Loop Euclidean
Lie algebras Lie group–Lie algebra correspondence Exponential map Adjoint representation Killing form Index Simple Lie algebra Loop algebra Affine Lie algebra
Semisimple Lie algebra Dynkin diagrams Cartan subalgebra Root system Weyl group Real form Complexification Split Lie algebra Compact Lie algebra
Representation theory Lie group representation Lie algebra representation Representation theory of semisimple Lie algebras Representations of classical Lie groups Theorem of the highest weight Borel–Weil–Bott theorem
Lie groups inphysics Particle physics and representation theory Lorentz group representations Poincaré group representations Galilean group representations
Scientists Sophus Lie Henri Poincaré Wilhelm Killing Élie Cartan Hermann Weyl Claude Chevalley Harish-Chandra Armand Borel
Glossary Table of Lie groups
v t e

Inmathematics, asymmetric space is aRiemannian manifold (or more generally, apseudo-Riemannian manifold) whose group of isometries contains aninversion symmetry about every point. This can be studied with the tools ofRiemannian geometry, leading to consequences in the theory ofholonomy; or algebraically throughLie theory, which allowedCartan to give a complete classification. Symmetric spaces commonly occur indifferential geometry,representation theory andharmonic analysis.

In geometric terms, a complete, simply connected Riemannian manifold is a symmetric space if and only if its curvature tensor is invariant under parallel transport. More generally, a Riemannian manifold (M,g) is said to be symmetric if and only if, for each pointp ofM, there exists an isometry ofM fixingp and acting on the tangent space${\displaystyle T_{p}M}$ as minus the identity (every symmetric space iscomplete, since any geodesic can be extended indefinitely via symmetries about the endpoints). Both descriptions can also naturally be extended to the setting ofpseudo-Riemannian manifolds.

From the point of view of Lie theory, a symmetric space is the quotientG / H of a connectedLie groupG by a Lie subgroupH that is (a connected component of) the invariant group of aninvolution ofG. This definition includes more than the Riemannian definition, and reduces to it whenH is compact.

Riemannian symmetric spaces arise in a wide variety of situations in both mathematics and physics. Their central role in the theory of holonomy was discovered byMarcel Berger. They are important objects of study in representation theory and harmonic analysis as well as in differential geometry.

LetM be a connected Riemannian manifold andp a point ofM. A diffeomorphismf of a neighborhood ofp is said to be ageodesic symmetry if it fixes the pointp and reverses geodesics through that point, i.e. ifγ is a geodesic with${\displaystyle \gamma (0)=p}$ then${\displaystyle f(\gamma (t))=\gamma (-t).}$ It follows that the derivative of the mapf atp is minus the identity map on thetangent space ofp. On a general Riemannian manifold,f need not be isometric, nor can it be extended, in general, from a neighbourhood ofp to all ofM.

M is said to belocally Riemannian symmetric if its geodesic symmetries are in fact isometric. This is equivalent to the vanishing of the covariant derivative of the curvature tensor. A locally symmetric space is said to be a(globally) symmetric space if in addition its geodesic symmetries can be extended to isometries on all ofM.

TheCartan–Ambrose–Hicks theorem implies thatM is locally Riemannian symmetricif and only if its curvature tensor iscovariantly constant, and furthermore that everysimply connected,complete locally Riemannian symmetric space is actually Riemannian symmetric.

Every Riemannian symmetric spaceM is complete and Riemannianhomogeneous (meaning that the isometry group ofM acts transitively onM). In fact, already the identity component of the isometry group acts transitively onM (becauseM is connected).

Locally Riemannian symmetric spaces that are not Riemannian symmetric may be constructed as quotients of Riemannian symmetric spaces by discrete groups of isometries with no fixed points, and as open subsets of (locally) Riemannian symmetric spaces.

Basic examples of Riemannian symmetric spaces areEuclidean space,spheres,projective spaces, andhyperbolic spaces, each with their standard Riemannian metrics. More examples are provided by compact, semi-simpleLie groups equipped with a bi-invariant Riemannian metric.

Every compactRiemann surface of genus greater than 1 (with its usual metric of constant curvature −1) is a locally symmetric space but not a symmetric space.

Everylens space is locally symmetric but not symmetric, with the exception of${\displaystyle L(2,1)}$, which is symmetric. The lens spaces are quotients of the 3-sphere by a discrete isometry that has no fixed points.

An example of a non-Riemannian symmetric space isanti-de Sitter space.

LetG be a connectedLie group. Then asymmetric space forG is a homogeneous spaceG / H where the stabilizerH of a typical point is an open subgroup of the fixed point set of aninvolutionσ in Aut(G). Thusσ is an automorphism ofG withσ² = id_G andH is an open subgroup of the invariant set

${\displaystyle G^{\sigma }=\{g\in G:\sigma (g)=g\}.}$

BecauseH is open, it is a union of components ofG^σ (including, of course, the identity component).

As an automorphism ofG,σ fixes the identity element, and hence, by differentiating at the identity, it induces an automorphism of the Lie algebra${\displaystyle \mathfrak{g}}$ ofG, also denoted byσ, whose square is the identity. It follows that the eigenvalues ofσ are ±1. The +1 eigenspace is the Lie algebra${\displaystyle \mathfrak{h}}$ ofH (since this is the Lie algebra ofG^σ), and the −1 eigenspace will be denoted${\displaystyle \mathfrak{m}}$. Sinceσ is an automorphism of${\displaystyle \mathfrak{g}}$, this gives adirect sum decomposition

${\displaystyle \mathfrak{g}=\mathfrak{h}\oplus \mathfrak{m}}$

with

${\displaystyle [\mathfrak{h},\mathfrak{h}]\subset \mathfrak{h},[\mathfrak{h},\mathfrak{m}]\subset \mathfrak{m},[\mathfrak{m},\mathfrak{m}]\subset \mathfrak{h}.}$

The first condition is automatic for any homogeneous space: it just says the infinitesimal stabilizer${\displaystyle \mathfrak{h}}$ is a Lie subalgebra of${\displaystyle \mathfrak{g}}$. The second condition means that${\displaystyle \mathfrak{m}}$ is an${\displaystyle \mathfrak{h}}$-invariant complement to${\displaystyle \mathfrak{h}}$ in${\displaystyle \mathfrak{g}}$. Thus any symmetric space is areductive homogeneous space, but there are many reductive homogeneous spaces which are not symmetric spaces. The key feature of symmetric spaces is the third condition that${\displaystyle \mathfrak{m}}$ brackets into${\displaystyle \mathfrak{h}}$.

Conversely, given any Lie algebra${\displaystyle \mathfrak{g}}$ with a direct sum decomposition satisfying these three conditions, the linear mapσ, equal to the identity on${\displaystyle \mathfrak{h}}$ and minus the identity on${\displaystyle \mathfrak{m}}$, is an involutive automorphism.

IfM is a Riemannian symmetric space, the identity componentG of the isometry group ofM is aLie group acting transitively onM (that is,M is Riemannian homogeneous). Therefore, if we fix some pointp ofM,M is diffeomorphic to the quotientG/K, whereK denotes theisotropy group of the action ofG onM atp. By differentiating the action atp we obtain an isometric action ofK on T_pM. This action is faithful (e.g., by a theorem of Kostant, any isometry in the identity component is determined by its1-jet at any point) and soK is a subgroup of the orthogonal group of T_pM, hence compact. Moreover, if we denote bys_p: M → M the geodesic symmetry ofM atp, the map

${\displaystyle \sigma :G\to G,h\mapsto s_p\circ h\circ s_p}$

is aninvolutive Lie groupautomorphism such that the isotropy groupK is contained between the fixed point group${\displaystyle G^{\sigma }}$ and its identity component (hence an open subgroup)${\displaystyle (G^{\sigma }{)}_o,}$ see the definition and following proposition on page 209, chapter IV, section 3 in Helgason's Differential Geometry, Lie Groups, and Symmetric Spaces for further information.

To summarize,M is a symmetric spaceG / K with a compact isotropy groupK. Conversely, symmetric spaces with compact isotropy group are Riemannian symmetric spaces, although not necessarily in a unique way. To obtain a Riemannian symmetric space structure we need to fix aK-invariant inner product on the tangent space toG / K at the identity coseteK: such an inner product always exists by averaging, sinceK is compact, and by acting withG, we obtain aG-invariant Riemannian metricg onG / K.

To show thatG / K is Riemannian symmetric, consider any pointp =hK (a coset ofK, whereh ∈G) and define

${\displaystyle s_p:M\to M,h^{\prime }K\mapsto h\sigma (h^{-1}{h}^{\prime })K}$

whereσ is the involution ofG fixingK. Then one can check thats_p is an isometry with (clearly)s_p(p) =p and (by differentiating) ds_p equal to minus the identity on T_pM. Thuss_p is a geodesic symmetry and, sincep was arbitrary,M is a Riemannian symmetric space.

If one starts with a Riemannian symmetric spaceM, and then performs these two constructions in sequence, then the Riemannian symmetric space yielded is isometric to the original one. This shows that the "algebraic data" (G,K,σ,g) completely describe the structure ofM.

The algebraic description of Riemannian symmetric spaces enabledÉlie Cartan to obtain a complete classification of them in 1926.

For a given Riemannian symmetric spaceM let (G,K,σ,g) be the algebraic data associated to it. To classify the possible isometry classes ofM, first note that theuniversal cover of a Riemannian symmetric space is again Riemannian symmetric, and the covering map is described by dividing the connected isometry groupG of the covering by a subgroup of its center. Therefore, we may suppose without loss of generality thatM is simply connected. (This impliesK is connected by thelong exact sequence of a fibration, becauseG is connected by assumption.)

A simply connected Riemannian symmetric space is said to beirreducible if it is not the product of two or more Riemannian symmetric spaces. It can then be shown that any simply connected Riemannian symmetric space is a Riemannian product of irreducible ones. Therefore, we may further restrict ourselves to classifying the irreducible, simply connected Riemannian symmetric spaces.

The next step is to show that any irreducible, simply connected Riemannian symmetric spaceM is of one of the following three types:

1. Euclidean type:M has vanishing curvature, and is therefore isometric to aEuclidean space.
2. Compact type:M has nonnegative (but not identically zero)sectional curvature.
3. Non-compact type:M has nonpositive (but not identically zero) sectional curvature.

A more refined invariant is therank, which is the maximum dimension of a subspace of the tangent space (to any point) on which the curvature is identically zero. The rank is always at least one, with equality if the sectional curvature is positive or negative. If the curvature is positive, the space is of compact type, and if negative, it is of noncompact type. The spaces of Euclidean type have rank equal to their dimension and are isometric to a Euclidean space of that dimension. Therefore, it remains to classify the irreducible, simply connected Riemannian symmetric spaces of compact and non-compact type. In both cases there are two classes.

A.G is a (real) simple Lie group;

B.G is either the product of a compact simple Lie group with itself (compact type), or a complexification of such a Lie group (non-compact type).

The examples in class B are completely described by the classification ofsimple Lie groups. For compact type,M is a compact simply connected simple Lie group,G isM×M andK is the diagonal subgroup. For non-compact type,G is a simply connected complex simple Lie group andK is its maximal compact subgroup. In both cases, the rank is therank ofG.

The compact simply connected Lie groups are the universal covers of the classical Lie groups SO(n), SU(n), Sp(n) and the fiveexceptional Lie groupsE₆,E₇,E₈,F₄,G₂.

The examples of class A are completely described by the classification of noncompact simply connected real simple Lie groups. For non-compact type,G is such a group andK is its maximal compact subgroup. Each such example has a corresponding example of compact type, by considering a maximal compact subgroup of the complexification ofG that containsK. More directly, the examples of compact type are classified by involutive automorphisms of compact simply connected simple Lie groupsG (up to conjugation). Such involutions extend to involutions of the complexification ofG, and these in turn classify non-compact real forms ofG.

In both class A and class B there is thus a correspondence between symmetric spaces of compact type and non-compact type. This is known as duality for Riemannian symmetric spaces.

Specializing to the Riemannian symmetric spaces of class A and compact type, Cartan found that there are the following seven infinite series and twelve exceptional Riemannian symmetric spacesG / K. They are here given in terms ofG andK, together with a geometric interpretation, if readily available. The labelling of these spaces is the one given by Cartan.

Label	G	K	Dimension	Rank	Geometric interpretation
AI	${\displaystyle \mathrm{S}\mathrm{U}(n)}$	${\displaystyle \mathrm{S}\mathrm{O}(n)}$	${\displaystyle (n-1)(n+2)/2}$	${\displaystyle n-1}$	Space of real structures on${\displaystyle {\mathbb{C}}^n}$ that leave the complex determinant invariant
AII	${\displaystyle \mathrm{S}\mathrm{U}(2n)}$	${\displaystyle \mathrm{S}\mathrm{p}(n)}$	${\displaystyle (n-1)(2n+1)}$	${\displaystyle n-1}$	Space of quaternionic structures on${\displaystyle {\mathbb{C}}^{2n}}$ compatible with the Hermitian metric
AIII	${\displaystyle \mathrm{S}\mathrm{U}(p+q)}$	${\displaystyle \mathrm{S}(\mathrm{U}(p)\times \mathrm{U}(q))}$	${\displaystyle 2pq}$	${\displaystyle min(p,q)}$	Grassmannian of complexp-dimensional subspaces of${\displaystyle {\mathbb{C}}^{p+q}}$
BDI	${\displaystyle \mathrm{S}\mathrm{O}(p+q)}$	${\displaystyle \mathrm{S}\mathrm{O}(p)\times \mathrm{S}\mathrm{O}(q)}$	${\displaystyle pq}$	${\displaystyle min(p,q)}$	Grassmannian of oriented realp-dimensional subspaces of${\displaystyle {\mathbb{R}}^{p+q}}$
DIII	${\displaystyle \mathrm{S}\mathrm{O}(2n)}$	${\displaystyle \mathrm{U}(n)}$	${\displaystyle n(n-1)}$	${\displaystyle [n/2]}$	Space of orthogonal complex structures on${\displaystyle {\mathbb{R}}^{2n}}$
CI	${\displaystyle \mathrm{S}\mathrm{p}(n)}$	${\displaystyle \mathrm{U}(n)}$	${\displaystyle n(n+1)}$	${\displaystyle n}$	Space of complex structures on${\displaystyle {\mathbb{H}}^n}$ compatible with the inner product
CII	${\displaystyle \mathrm{S}\mathrm{p}(p+q)}$	${\displaystyle \mathrm{S}\mathrm{p}(p)\times \mathrm{S}\mathrm{p}(q)}$	${\displaystyle 4pq}$	${\displaystyle min(p,q)}$	Grassmannian of quaternionicp-dimensional subspaces of${\displaystyle {\mathbb{H}}^{p+q}}$
EI	${\displaystyle E_6}$	${\displaystyle \mathrm{S}\mathrm{p}(4)/\{\pm I\}}$	42	6
EII	${\displaystyle E_6}$	${\displaystyle \mathrm{S}\mathrm{U}(6)\cdot \mathrm{S}\mathrm{U}(2)}$	40	4	Space of symmetric subspaces of${\displaystyle (\mathbb{C}\otimes \mathbb{O})P^2}$ isometric to${\displaystyle (\mathbb{C}\otimes \mathbb{H})P^2}$
EIII	${\displaystyle E_6}$	${\displaystyle \mathrm{S}\mathrm{O}(10)\cdot \mathrm{S}\mathrm{O}(2)}$	32	2	ComplexifiedCayley projective plane${\displaystyle (\mathbb{C}\otimes \mathbb{O})P^2}$
EIV	${\displaystyle E_6}$	${\displaystyle F_4}$	26	2	Space of symmetric subspaces of${\displaystyle (\mathbb{C}\otimes \mathbb{O})P^2}$ isometric to${\displaystyle {\mathbb{O}\mathbb{P}}^2}$
EV	${\displaystyle E_7}$	${\displaystyle \mathrm{S}\mathrm{U}(8)/\{\pm I\}}$	70	7
EVI	${\displaystyle E_7}$	${\displaystyle \mathrm{S}\mathrm{O}(12)\cdot \mathrm{S}\mathrm{U}(2)}$	64	4	Rosenfeld projective plane${\displaystyle (\mathbb{H}\otimes \mathbb{O})P^2}$ over${\displaystyle \mathbb{H}\otimes \mathbb{O}}$
EVII	${\displaystyle E_7}$	${\displaystyle E_6\cdot \mathrm{S}\mathrm{O}(2)}$	54	3	Space of symmetric subspaces of${\displaystyle (\mathbb{H}\otimes \mathbb{O})P^2}$ isomorphic to${\displaystyle (\mathbb{C}\otimes \mathbb{O})P^2}$
EVIII	${\displaystyle E_8}$	${\displaystyle \mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}(16)/\{\pm \mathrm{v}\mathrm{o}\mathrm{l}\}}$	128	8	Rosenfeld projective plane${\displaystyle (\mathbb{O}\otimes \mathbb{O})P^2}$
EIX	${\displaystyle E_8}$	${\displaystyle E_7\cdot \mathrm{S}\mathrm{U}(2)}$	112	4	Space of symmetric subspaces of${\displaystyle (\mathbb{O}\otimes \mathbb{O})P^2}$ isomorphic to${\displaystyle (\mathbb{H}\otimes \mathbb{O})P^2}$
FI	${\displaystyle F_4}$	${\displaystyle \mathrm{S}\mathrm{p}(3)\cdot \mathrm{S}\mathrm{U}(2)}$	28	4	Space of symmetric subspaces of${\displaystyle \mathbb{O}{P}^2}$ isomorphic to${\displaystyle \mathbb{H}{P}^2}$
FII	${\displaystyle F_4}$	${\displaystyle \mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}(9)}$	16	1	Cayley projective plane${\displaystyle \mathbb{O}{P}^2}$
G	${\displaystyle G_2}$	${\displaystyle \mathrm{S}\mathrm{O}(4)}$	8	2	Space of subalgebras of theoctonion algebra${\displaystyle \mathbb{O}}$ which are isomorphic to thequaternion algebra${\displaystyle \mathbb{H}}$

A more modern classification (Huang & Leung 2010) uniformly classifies the Riemannian symmetric spaces, both compact and non-compact, via aFreudenthal magic square construction. The irreducible compact Riemannian symmetric spaces are, up to finite covers, either a compact simple Lie group, a Grassmannian, aLagrangian Grassmannian, or adouble Lagrangian Grassmannian of subspaces of${\displaystyle (\mathbf{A}\otimes \mathbf{B}{)}^n,}$ for normed division algebrasA andB. A similar construction produces the irreducible non-compact Riemannian symmetric spaces.

An important class of symmetric spaces generalizing the Riemannian symmetric spaces arepseudo-Riemannian symmetric spaces, in which the Riemannian metric is replaced by apseudo-Riemannian metric (nondegenerate instead of positive definite on each tangent space). In particular,Lorentzian symmetric spaces, i.e.,n dimensional pseudo-Riemannian symmetric spaces of signature (n − 1,1), are important ingeneral relativity, the most notable examples beingMinkowski space,De Sitter space andanti-de Sitter space (with zero, positive and negative curvature respectively). De Sitter space of dimensionn may be identified with the 1-sheeted hyperboloid in a Minkowski space of dimensionn + 1.

Symmetric and locally symmetric spaces in general can be regarded as affine symmetric spaces. IfM =G / H is a symmetric space, then Nomizu showed that there is aG-invariant torsion-freeaffine connection (i.e. an affine connection whosetorsion tensor vanishes) onM whosecurvature isparallel. Conversely a manifold with such a connection is locally symmetric (i.e., itsuniversal cover is a symmetric space). Such manifolds can also be described as those affine manifolds whose geodesic symmetries are all globally defined affine diffeomorphisms, generalizing the Riemannian and pseudo-Riemannian case.

The classification of Riemannian symmetric spaces does not extend readily to the general case for the simple reason that there is no general splitting of a symmetric space into a product of irreducibles. Here a symmetric spaceG / H with Lie algebra

${\displaystyle \mathfrak{g}=\mathfrak{h}\oplus \mathfrak{m}}$

is said to be irreducible if${\displaystyle \mathfrak{m}}$ is anirreducible representation of${\displaystyle \mathfrak{h}}$. Since${\displaystyle \mathfrak{h}}$ is not semisimple (or even reductive) in general, it can haveindecomposable representations which are not irreducible.

However, the irreducible symmetric spaces can be classified. As shown byKatsumi Nomizu, there is a dichotomy: an irreducible symmetric spaceG / H is either flat (i.e., an affine space) or${\displaystyle \mathfrak{g}}$ is semisimple. This is the analogue of the Riemannian dichotomy between Euclidean spaces and those of compact or noncompact type, and it motivated M. Berger to classify semisimple symmetric spaces (i.e., those with${\displaystyle \mathfrak{g}}$ semisimple) and determine which of these are irreducible. The latter question is more subtle than in the Riemannian case: even if${\displaystyle \mathfrak{g}}$ is simple,G / H might not be irreducible.

As in the Riemannian case there are semisimple symmetric spaces withG =H ×H. Any semisimple symmetric space is a product of symmetric spaces of this form with symmetric spaces such that${\displaystyle \mathfrak{g}}$ is simple. It remains to describe the latter case. For this, one needs to classify involutionsσ of a (real) simple Lie algebra${\displaystyle \mathfrak{g}}$. If${\displaystyle {\mathfrak{g}}^c}$ is not simple, then${\displaystyle \mathfrak{g}}$ is a complex simple Lie algebra, and the corresponding symmetric spaces have the formG / H, whereH is a real form ofG: these are the analogues of the Riemannian symmetric spacesG / K withG a complex simple Lie group, andK a maximal compact subgroup.

Thus we may assume${\displaystyle {\mathfrak{g}}^c}$ is simple. The real subalgebra${\displaystyle \mathfrak{g}}$ may be viewed as the fixed point set of a complexantilinear involutionτ of${\displaystyle {\mathfrak{g}}^c}$, whileσ extends to a complex antilinear involution of${\displaystyle {\mathfrak{g}}^c}$ commuting withτ and hence also a complex linear involutionσ∘τ.

The classification therefore reduces to the classification of commuting pairs of antilinear involutions of a complex Lie algebra. The compositeσ∘τ determines a complex symmetric space, whileτ determines a real form. From this it is easy to construct tables of symmetric spaces for any given${\displaystyle {\mathfrak{g}}^c}$, and furthermore, there is an obvious duality given by exchangingσ andτ. This extends the compact/non-compact duality from the Riemannian case, where eitherσ orτ is aCartan involution, i.e., its fixed point set is a maximal compact subalgebra.

The following table indexes the real symmetric spaces by complex symmetric spaces and real forms, for each classical and exceptional complex simple Lie group.

For exceptional simple Lie groups, the Riemannian case is included explicitly below, by allowingσ to be the identity involution (indicated by a dash). In the above tables this is implicitly covered by the casekl = 0.

In the 1950sAtle Selberg extended Cartan's definition of symmetric space to that ofweakly symmetric Riemannian space, or in current terminologyweakly symmetric space. These are defined as Riemannian manifoldsM with a transitive connected Lie group of isometriesG and an isometryσ normalisingG such that givenx,y inM there is an isometrys inG such thatsx =σy andsy =σx. (Selberg's assumption thatσ² should be an element ofG was later shown to be unnecessary byErnest Vinberg.) Selberg proved that weakly symmetric spaces give rise toGelfand pairs, so that in particular theunitary representation ofG onL²(M) is multiplicity free.

Selberg's definition can also be phrased equivalently in terms of a generalization of geodesic symmetry. It is required that for every pointx inM and tangent vectorX atx, there is an isometrys ofM, depending onx andX, such that

• s fixesx;
• the derivative ofs atx sendsX to −X.

Whens is independent ofX,M is a symmetric space.

An account of weakly symmetric spaces and their classification by Akhiezer and Vinberg, based on the classification of periodic automorphisms of complexsemisimple Lie algebras, is given inWolf (2007).

Some properties and forms of symmetric spaces can be noted.

Themetric tensor on the Riemannian manifoldM can be lifted to a scalar product onG by combining it with theKilling form. This is done by defining

Here,${\displaystyle ⟨\cdot ,\cdot {⟩}_p}$ is the Riemannian metric defined on${\displaystyle T_{p}M}$, and${\displaystyle B(X,Y)=\mathrm{trace}(\mathrm{ad}X\circ \mathrm{ad}Y)}$ is theKilling form. The minus sign appears because the Killing form is negative-definite on${\displaystyle \mathfrak{h};}$ this makes${\displaystyle ⟨\cdot ,\cdot {⟩}_{\mathfrak{g}}}$ positive-definite.

The tangent space${\displaystyle \mathfrak{m}}$ can be further factored into eigenspaces classified by the Killing form.^[1] This is accomplished by defining an adjoint map${\displaystyle \mathfrak{m}\to \mathfrak{m}}$ taking${\displaystyle Y\mapsto Y^{\mathrm{\#}}}$ as

${\displaystyle ⟨X,Y^{\mathrm{\#}}⟩=B(X,Y)}$

where${\displaystyle ⟨\cdot ,\cdot ⟩}$ is the Riemannian metric on${\displaystyle \mathfrak{m}}$ and${\displaystyle B(\cdot ,\cdot )}$ is the Killing form. This map is sometimes called thegeneralized transpose, as corresponds to the transpose for the orthogonal groups and the Hermitian conjugate for the unitary groups. It is a linear functional, and it is self-adjoint, and so one concludes that there is an orthonormal basis${\displaystyle Y_1,\dots ,Y_n}$ of${\displaystyle \mathfrak{m}}$ with

${\displaystyle Y_i^{\mathrm{\#}}={\lambda }_i{Y}_i}$

These are orthogonal with respect to the metric, in that

${\displaystyle ⟨Y_i^{\mathrm{\#}},Y_j⟩={\lambda }_i⟨Y_i,Y_j⟩=B(Y_i,Y_j)=⟨Y_j^{\mathrm{\#}},Y_i⟩={\lambda }_j⟨Y_j,Y_i⟩}$

since the Killing form is symmetric. This factorizes${\displaystyle \mathfrak{m}}$ into eigenspaces

${\displaystyle \mathfrak{m}={\mathfrak{m}}_1\oplus \cdots \oplus {\mathfrak{m}}_d}$

with

${\displaystyle [{\mathfrak{m}}_i,{\mathfrak{m}}_j]=0}$

for${\displaystyle i\ne j}$. For the case of${\displaystyle \mathfrak{g}}$ semisimple, so that the Killing form is non-degenerate, the metric likewise factorizes:

${\displaystyle ⟨\cdot ,\cdot ⟩=\frac{1}{{\lambda }_1}{B|}_{{\mathfrak{m}}_1}+\cdots +\frac{1}{{\lambda }_d}{B|}_{{\mathfrak{m}}_d}}$

In certain practical applications, this factorization can be interpreted as the spectrum of operators,e.g. the spectrum of the hydrogen atom, with the eigenvalues of the Killing form corresponding to different values of the angular momentum of an orbital (i.e. the Killing form being aCasimir operator that can classify the different representations under which different orbitals transform.)

Classification of symmetric spaces proceeds based on whether or not the Killing form is definite.

If the identity component of theholonomy group of a Riemannian manifold at a point acts irreducibly on the tangent space, then either the manifold is a locally Riemannian symmetric space, or it is in one of7 families.

A Riemannian symmetric space that is additionally equipped with a parallel complex structure compatible with the Riemannian metric is called aHermitian symmetric space. Some examples are complex vector spaces and complex projective spaces, both with their usual Riemannian metric, and the complex unit balls with suitable metrics so that they become complete and Riemannian symmetric.

An irreducible symmetric spaceG / K is Hermitian if and only ifK contains a central circle. A quarter turn by this circle acts as multiplication byi on the tangent space at the identity coset. Thus the Hermitian symmetric spaces are easily read off of the classification. In both the compact and the non-compact cases it turns out that there are four infinite series, namely AIII, BDI withp = 2, DIII and CI, and two exceptional spaces, namely EIII and EVII. The non-compact Hermitian symmetric spaces can be realized as bounded symmetric domains in complex vector spaces.

A Riemannian symmetric space that is additionally equipped with a parallel subbundle of End(TM) isomorphic to the imaginary quaternions at each point, and compatible with the Riemannian metric, is calledquaternion-Kähler symmetric space.

An irreducible symmetric spaceG / K is quaternion-Kähler if and only if isotropy representation ofK contains an Sp(1) summand acting like theunit quaternions on aquaternionic vector space. Thus the quaternion-Kähler symmetric spaces are easily read off from the classification. In both the compact and the non-compact cases it turns out that there is exactly one for each complex simple Lie group, namely AI withp = 2 orq = 2 (these are isomorphic), BDI withp = 4 orq = 4, CII withp = 1 orq = 1, EII, EVI, EIX, FI and G.

In theBott periodicity theorem, theloop spaces of the stableorthogonal group can be interpreted as reductive symmetric spaces.

• Orthogonal symmetric Lie algebra
• Relative root system
• Satake diagram
• Cartan involution

• Akhiezer, D. N.;Vinberg, E. B. (1999), "Weakly symmetric spaces and spherical varieties",Transf. Groups,4:3–24,doi:10.1007/BF01236659
• van den Ban, E. P.; Flensted-Jensen, M.; Schlichtkrull, H. (1997),Harmonic analysis on semisimple symmetric spaces: A survey of some general results, in Representation Theory and Automorphic Forms: Instructional Conference, International Centre for Mathematical Sciences, March 1996, Edinburgh, Scotland, American Mathematical Society,ISBN 978-0-8218-0609-8
• Berger, Marcel (1957), "Les espaces symétriques noncompacts",Annales Scientifiques de l'École Normale Supérieure,74 (2):85–177,doi:10.24033/asens.1054
• Besse, Arthur Lancelot (1987),Einstein Manifolds, Springer-Verlag,ISBN 0-387-15279-2 Contains a compact introduction and many tables.
• Borel, Armand (2001),Essays in the History of Lie Groups and Algebraic Groups, American Mathematical Society,ISBN 0-8218-0288-7
• Cartan, Élie (1926), "Sur une classe remarquable d'espaces de Riemann, I",Bulletin de la Société Mathématique de France,54:214–216,doi:10.24033/bsmf.1105
• Cartan, Élie (1927), "Sur une classe remarquable d'espaces de Riemann, II",Bulletin de la Société Mathématique de France,55:114–134,doi:10.24033/bsmf.1113
• Flensted-Jensen, Mogens (1986),Analysis on Non-Riemannian Symmetric Spaces, CBMS Regional Conference, American Mathematical Society,ISBN 978-0-8218-0711-8
• Helgason, Sigurdur (1978),Differential geometry, Lie groups and symmetric spaces, Academic Press,ISBN 0-12-338460-5 The standard book on Riemannian symmetric spaces.
• Helgason, Sigurdur (1984),Groups and Geometric Analysis: Integral Geometry, Invariant Differential Operators, and Spherical Functions, Academic Press,ISBN 0-12-338301-3
• Huang, Yongdong; Leung, Naichung Conan (2010)."A uniform description of compact symmetric spaces as Grassmannians using the magic square"(PDF).Mathematische Annalen.350 (1):79–106.doi:10.1007/s00208-010-0549-8.
• Kobayashi, Shoshichi; Nomizu, Katsumi (1996),Foundations of Differential Geometry, Volume II, Wiley Classics Library edition,ISBN 0-471-15732-5 Chapter XI contains a good introduction to Riemannian symmetric spaces.
• Loos, Ottmar (1969),Symmetric spaces I: General Theory, Benjamin
• Loos, Ottmar (1969),Symmetric spaces II: Compact Spaces and Classification, Benjamin
• Nomizu, K. (1954),"Invariant affine connections on homogeneous spaces",Amer. J. Math.,76 (1):33–65,doi:10.2307/2372398,JSTOR 2372398
• Selberg, Atle (1956), "Harmonic analysis and discontinuous groups in weakly symmetric riemannian spaces, with applications to Dirichlet series",J. Indian Math. Society,20:47–87
• Wolf, Joseph A. (1999),Spaces of constant curvature (5th ed.), McGraw–Hill
• Wolf, Joseph A. (2007),Harmonic Analysis on Commutative Spaces, American Mathematical Society,ISBN 978-0-8218-4289-8

• Differential geometry
• Riemannian geometry
• Lie groups
• Homogeneous spaces

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