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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

In mathematics, asimple Lie group is aconnectednon-abelianLie groupG which does not have nontrivial connectednormal subgroups. The list of simple Lie groups can be used to read off the list ofsimple Lie algebras andRiemannian symmetric spaces.

Together with the commutative Lie group of the real numbers,${\displaystyle \mathbb{R}}$, and that of the unit-magnitude complex numbers,U(1) (the unit circle), simple Lie groups give the atomic "building blocks" that make up all (finite-dimensional) connected Lie groups via the operation ofgroup extension. Many commonly encountered Lie groups are either simple or 'close' to being simple: for example, the so-called "special linear group" SL(n,${\displaystyle \mathbb{R}}$) ofn byn matrices with determinant equal to 1 is simple for all oddn > 1, when it is isomorphic to theprojective special linear group.

The first classification of simple Lie groups was byWilhelm Killing, and this work was later perfected byÉlie Cartan. The final classification is often referred to as Killing-Cartan classification.

Unfortunately, there is no universally accepted definition of a simple Lie group. In particular, it is not always defined as a Lie group that issimple as an abstract group. Authors differ on whether a simple Lie group has to be connected, or on whether it is allowed to have a non-trivial center, or on whether${\displaystyle \mathbb{R}}$ is a simple Lie group.

The most common definition is that a Lie group is simple if it is connected, non-abelian, and every closedconnected normal subgroup is either the identity or the whole group. In particular, simple groups are allowed to have a non-trivial center, but${\displaystyle \mathbb{R}}$ is not simple.

In this article the connected simple Lie groups with trivial center are listed. Once these are known, the ones with non-trivial center are easy to list as follows. Any simple Lie group with trivial center has auniversal cover whose center is thefundamental group of the simple Lie group. The corresponding simple Lie groups with non-trivial center can be obtained as quotients of this universal cover by a subgroup of the center.

An equivalent definition of a simple Lie group follows from theLie correspondence: A connected Lie group is simple if itsLie algebra issimple. An important technical point is that a simple Lie group may containdiscrete normal subgroups. For this reason, the definition of a simple Lie group is not equivalent to the definition of a Lie group that issimple as an abstract group.

Simple Lie groups include manyclassical Lie groups, which provide a group-theoretic underpinning forspherical geometry,projective geometry and related geometries in the sense ofFelix Klein'sErlangen program. It emerged in the course ofclassification of simple Lie groups that there exist also severalexceptional possibilities not corresponding to any familiar geometry. Theseexceptional groups account for many special examples and configurations in other branches of mathematics, as well as contemporarytheoretical physics.

As a counterexample, thegeneral linear group is neither simple, norsemisimple. This is because multiples of the identity form a nontrivial normal subgroup, thus evading the definition. Equivalently, the correspondingLie algebra has a degenerateKilling form, because multiples of the identity map to the zero element of the algebra. Thus, the corresponding Lie algebra is also neither simple nor semisimple. Another counter-example are thespecial orthogonal groups in even dimension. These have the matrix${\displaystyle -I}$ in thecenter, and this element is path-connected to the identity element, and so these groups evade the definition. Both of these arereductive groups.

Asemisimple Lie group is a connected Lie group so that its onlyclosedconnectedabeliannormal subgroup is the trivial subgroup. Every simple Lie group is semisimple. More generally, any product of simple Lie groups is semisimple, and any quotient of a semisimple Lie group by a closed subgroup is semisimple. Every semisimple Lie group can be formed by taking a product of simple Lie groups and quotienting by a subgroup of its center. In other words, every semisimple Lie group is acentral product of simple Lie groups. The semisimple Lie groups are exactly the Lie groups whose Lie algebras aresemisimple Lie algebras.

TheLie algebra of a simple Lie group is a simple Lie algebra. This is a one-to-one correspondence between connected simple Lie groups withtrivial center and simple Lie algebras of dimension greater than 1. (Authors differ on whether the one-dimensional Lie algebra should be counted as simple.)

Over the complex numbers the semisimple Lie algebras are classified by theirDynkin diagrams, of types "ABCDEFG". IfL is a real simple Lie algebra, its complexification is a simple complex Lie algebra, unlessL is already the complexification of a Lie algebra, in which case the complexification ofL is a product of two copies ofL. This reduces the problem of classifying the real simple Lie algebras to that of finding all thereal forms of each complex simple Lie algebra (i.e., real Lie algebras whose complexification is the given complex Lie algebra). There are always at least 2 such forms: a split form and a compact form, and there are usually a few others. The different real forms correspond to the classes of automorphisms of order at most 2 of the complex Lie algebra.

Symmetric spaces are classified as follows.

First, the universal cover of a symmetric space is still symmetric, so we can reduce to the case of simply connected symmetric spaces. (For example, the universal cover of a real projective plane is a sphere.)

Second, the product of symmetric spaces is symmetric, so we may as well just classify the irreducible simply connected ones (where irreducible means they cannot be written as a product of smaller symmetric spaces).

The irreducible simply connected symmetric spaces are the real line, and exactly two symmetric spaces corresponding to eachnon-compact simple Lie groupG,one compact and one non-compact. The non-compact one is a cover of the quotient ofG by a maximal compact subgroupH, and the compact one is a cover of the quotient ofthe compact form ofG by the same subgroupH. This duality between compact and non-compact symmetric spaces is a generalization of the well known duality between spherical and hyperbolic geometry.

A symmetric space with a compatible complex structure is called Hermitian. The compact simply connected irreducible Hermitian symmetric spaces fall into 4 infinite families with 2 exceptional ones left over, and each has a non-compact dual. In addition the complex plane is also a Hermitian symmetric space; this gives the complete list of irreducible Hermitian symmetric spaces.

The four families are the types A III, B I and D I forp = 2, D III, and C I, and the two exceptional ones are types E III and E VII of complex dimensions 16 and 27.

${\displaystyle \mathbb{R}\mathbb{,}\mathbb{C}\mathbb{,}\mathbb{H}\mathbb{,}\mathbb{O}}$  stand for the real numbers, complex numbers,quaternions, andoctonions.

In the symbols such asE₆⁻²⁶ for the exceptional groups, the exponent −26 is the signature of an invariant symmetric bilinear form that is negative definite on the maximal compact subgroup. It is equal to the dimension of the group minus twice the dimension of a maximal compact subgroup.

The fundamental group listed in the table below is the fundamental group of the simple group with trivial center. Other simple groups with the same Lie algebra correspond to subgroups of this fundamental group (modulo the action of the outer automorphism group).

Simple Lie groups are fully classified. The classification is usually stated in several steps, namely:

• Classification of simple complex Lie algebras The classification of simple Lie algebras over the complex numbers byDynkin diagrams.
• Classification of simple real Lie algebras Each simple complex Lie algebra has severalreal forms, classified by additional decorations of its Dynkin diagram calledSatake diagrams, afterIchirô Satake.
• Classification of centerless simple Lie groups For every (real or complex) simple Lie algebra${\displaystyle \mathfrak{g}}$, there is a unique "centerless" simple Lie group${\displaystyle G}$ whose Lie algebra is${\displaystyle \mathfrak{g}}$ and which has trivialcenter.
• Classification of simple Lie groups

One can show that thefundamental group of any Lie group is a discretecommutative group. Given a (nontrivial) subgroup${\displaystyle K\subset {\pi }_1(G)}$ of the fundamental group of some Lie group${\displaystyle G}$, one can use the theory ofcovering spaces to construct a new group${\displaystyle {\tilde{G}}^K}$ with${\displaystyle K}$ in its center. Now any (real or complex) Lie group can be obtained by applying this construction to centerless Lie groups. Note that real Lie groups obtained this way might not be real forms of any complex group. A very important example of such a real group is themetaplectic group, which appears in infinite-dimensional representation theory and physics. When one takes for${\displaystyle K\subset {\pi }_1(G)}$ the full fundamental group, the resulting Lie group${\displaystyle {\tilde{G}}^{K={\pi }_1(G)}}$ is the universal cover of the centerless Lie group${\displaystyle G}$, and is simply connected. In particular, every (real or complex) Lie algebra also corresponds to a unique connected andsimply connected Lie group${\displaystyle \tilde{G}}$ with that Lie algebra, called the "simply connected Lie group" associated to${\displaystyle \mathfrak{g}.}$

Every simple complex Lie algebra has a unique real form whose corresponding centerless Lie group iscompact. It turns out that the simply connected Lie group in these cases is also compact. Compact Lie groups have a particularly tractable representation theory because of thePeter–Weyl theorem. Just like simple complex Lie algebras, centerless compact Lie groups are classified by Dynkin diagrams (first classified byWilhelm Killing andÉlie Cartan).

For the infinite (A, B, C, D) series of Dynkin diagrams, a connected compact Lie group associated to each Dynkin diagram can be explicitly described as a matrix group, with the corresponding centerless compact Lie group described as the quotient by a subgroup of scalar matrices. For those of type A and C we can find explicit matrix representations of the corresponding simply connected Lie group as matrix groups.

A_r has as its associated simply connected compact group thespecial unitary group,SU(r + 1) and as its associated centerless compact group the projective unitary groupPU(r + 1).

B_r has as its associated centerless compact groups the oddspecial orthogonal groups,SO(2r + 1). This group is not simply connected however: its universal (double) cover is thespin group.

C_r has as its associated simply connected group the group ofunitary symplectic matrices,Sp(r) and as its associated centerless group the Lie groupPSp(r) = Sp(r)/{I, −I} of projective unitary symplectic matrices. The symplectic groups have a double-cover by themetaplectic group.

D_r has as its associated compact group the evenspecial orthogonal groups,SO(2r) and as its associated centerless compact group the projective special orthogonal groupPSO(2r) = SO(2r)/{I, −I}. As with the B series, SO(2r) is not simply connected; its universal cover is again thespin group, but the latter again has a center (cf. its article).

The diagram D₂ is two isolated nodes, the same as A₁ ∪ A₁, and this coincidence corresponds to the covering map homomorphism from SU(2) × SU(2) to SO(4) given byquaternion multiplication; seequaternions and spatial rotation. Thus SO(4) is not a simple group. Also, the diagram D₃ is the same as A₃, corresponding to a covering map homomorphism from SU(4) to SO(6).

In addition to the four familiesA_i,B_i,C_i, andD_i above, there are five so-called exceptional Dynkin diagramsG₂,F₄,E₆,E₇, andE₈; these exceptional Dynkin diagrams also have associated simply connected and centerless compact groups. However, the groups associated to the exceptional families are more difficult to describe than those associated to the infinite families, largely because their descriptions make use ofexceptional objects. For example, the group associated to G₂ is the automorphism group of theoctonions, and the group associated to F₄ is the automorphism group of a certainAlbert algebra.

See also⁠${\displaystyle E_{7\frac{1}{2}}}$⁠.

	Dimension	Outer automorphism group	Dimension of symmetric space	Symmetric space	Remarks
${\displaystyle \mathbb{R}}$ (Abelian)	1	${\displaystyle {\mathbb{R}}^{\ast }}$	1	${\displaystyle \mathbb{R}}$	†

^† The group${\displaystyle \mathbb{R}}$ is not 'simple' as an abstract group, and according to most (but not all) definitions this is not a simple Lie group. Further, most authors do not count its Lie algebra as a simple Lie algebra. It is listed here so that the list of "irreducible simply connected symmetric spaces" is complete. Note that${\displaystyle \mathbb{R}}$ is the only such non-compact symmetric space without a compact dual (although it has a compact quotientS¹).

The following table lists some Lie groups with simple Lie algebras of small dimension. The groups on a given line all have the same Lie algebra. In the dimension 1 case, the groups are abelian and not simple.

Dim	Groups		Symmetric space	Compact dual	Rank	Dim
1	ℝ,S1 = U(1) = SO2(ℝ) = Spin(2)	Abelian	Real line		0	1
3	S3 = Sp(1) = SU(2)=Spin(3), SO3(ℝ) = PSU(2)	Compact
3	SL2(ℝ) = Sp2(ℝ), SO2,1(ℝ)	Split, Hermitian, hyperbolic	Hyperbolic plane${\displaystyle {\mathbb{H}}^2}$	SphereS2	1	2
6	SL2(ℂ) = Sp2(ℂ), SO3,1(ℝ), SO3(ℂ)	Complex	Hyperbolic space${\displaystyle {\mathbb{H}}^3}$	SphereS3	1	3
8	SL3(ℝ)	Split	Euclidean structures on${\displaystyle {\mathbb{R}}^3}$	Real structures on${\displaystyle {\mathbb{C}}^3}$	2	5
8	SU(3)	Compact
8	SU(1,2)	Hermitian, quasi-split, quaternionic	Complex hyperbolic plane	Complex projective plane	1	4
10	Sp(2) = Spin(5), SO5(ℝ)	Compact
10	SO4,1(ℝ), Sp2,2(ℝ)	Hyperbolic, quaternionic	Hyperbolic space${\displaystyle {\mathbb{H}}^4}$	SphereS4	1	4
10	SO3,2(ℝ), Sp4(ℝ)	Split, Hermitian	Siegel upper half space	Complex structures on${\displaystyle {\mathbb{H}}^2}$	2	6
14	G2	Compact
14	G2	Split, quaternionic	Non-division quaternionic subalgebras of non-division octonions	Quaternionic subalgebras of octonions	2	8
15	SU(4) = Spin(6), SO6(ℝ)	Compact
15	SL4(ℝ), SO3,3(ℝ)	Split	ℝ3 in ℝ3,3	GrassmannianG(3,3)	3	9
15	SU(3,1)	Hermitian	Complex hyperbolic space	Complex projective space	1	6
15	SU(2,2), SO4,2(ℝ)	Hermitian, quasi-split, quaternionic	ℝ2 in ℝ2,4	GrassmannianG(2,4)	2	8
15	SL2(ℍ), SO5,1(ℝ)	Hyperbolic	Hyperbolic space${\displaystyle {\mathbb{H}}^5}$	SphereS5	1	5
16	SL3(ℂ)	Complex		SU(3)	2	8
20	SO5(ℂ), Sp4(ℂ)	Complex		Spin5(ℝ)	2	10
21	SO7(ℝ)	Compact
21	SO6,1(ℝ)	Hyperbolic	Hyperbolic space${\displaystyle {\mathbb{H}}^6}$	SphereS6
21	SO5,2(ℝ)	Hermitian
21	SO4,3(ℝ)	Split, quaternionic
21	Sp(3)	Compact
21	Sp6(ℝ)	Split, hermitian
21	Sp4,2(ℝ)	Quaternionic
24	SU(5)	Compact
24	SL5(ℝ)	Split
24	SU4,1	Hermitian
24	SU3,2	Hermitian, quaternionic
28	SO8(ℝ)	Compact
28	SO7,1(ℝ)	Hyperbolic	Hyperbolic space${\displaystyle {\mathbb{H}}^7}$	SphereS7
28	SO6,2(ℝ)	Hermitian
28	SO5,3(ℝ)	Quasi-split
28	SO4,4(ℝ)	Split, quaternionic
28	SO∗8(ℝ)	Hermitian
28	G2(ℂ)	Complex
30	SL4(ℂ)	Complex

Asimply laced group is aLie group whoseDynkin diagram only contain simple links, and therefore all the nonzero roots of the corresponding Lie algebra have the same length. The A, D and E series groups are all simply laced, but no group of type B, C, F, or G is simply laced.

• Cartan matrix
• Coxeter matrix
• Weyl group
• Coxeter group
• Kac–Moody algebra
• Catastrophe theory
• Table of Lie groups
• Classification of low-dimensional real Lie algebras

• Jacobson, Nathan (1971).Exceptional Lie Algebras. CRC Press.ISBN 0-8247-1326-5.
• Fulton, William;Harris, Joe (2004).Representation Theory: A First Course. Springer.doi:10.1007/978-1-4612-0979-9.ISBN 978-1-4612-0979-9.

• Besse,Einstein manifoldsISBN 0-387-15279-2
• Helgason,Differential geometry, Lie groups, and symmetric spaces.ISBN 0-8218-2848-7
• Fuchs and Schweigert,Symmetries, Lie algebras, and representations: a graduate course for physicists. Cambridge University Press, 2003.ISBN 0-521-54119-0

• Lie groups
• Lie algebras

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