Lie groups and Lie 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 in physics 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

In mathematics, a simple Lie group is a connected 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 Lie algebras and Riemannian 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 "blocks" that make up all (finite-dimensional) connected Lie groups via the operation of group extension. Many commonly encountered Lie groups are either simple or 'close' to being simple: for example, the so-called "special linear group" SL(n) of n by n matrices with determinant equal to 1 is simple for all n > 1.

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

Definition

Unfortunately, there is no universally accepted definition of a simple Lie group. In particular, it is not always defined as a Lie group that is simple 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 closed connected 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 a universal cover, whose center is the fundamental 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.

Related ideas

Full classification

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 by Dynkin diagrams.
• Classification of simple real Lie algebras Each simple complex Lie algebra has several real forms, classified by additional decorations of its Dynkin diagram called Satake diagrams, after Ichirô 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 trivial center.
• Classification of simple Lie groups

One can show that the fundamental group of any Lie group is a discrete commutative 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 of covering 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 the metaplectic 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 and simply connected Lie group ${\displaystyle {\tilde {G}}}$ with that Lie algebra, called the "simply connected Lie group" associated to ${\displaystyle {\mathfrak {g}}.}$

Overview of the classification

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

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

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

D_r has as its associated compact group the even special orthogonal groups, SO(2r) and as its associated centerless compact group the projective special orthogonal group PSO(2r) = SO(2r)/{I, −I}. As with the B series, SO(2r) is not simply connected; its universal cover is again the spin 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 by quaternion multiplication; see quaternions 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 families A_i, B_i, C_i, and D_i above, there are five so-called exceptional Dynkin diagrams G₂, F₄, E₆, E₇, and E₈; 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 of exceptional objects. For example, the group associated to G₂ is the automorphism group of the octonions, and the group associated to F₄ is the automorphism group of a certain Albert algebra.

See also E_7+1⁄2.

List

Abelian

See also: Abelian group

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

Notes

^† 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 quotient S¹).

Simple Lie groups of small dimension

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	${\displaystyle \mathbb {R} }$, S1 = U(1) = SO2(${\displaystyle \mathbb {R} }$) = Spin(2)	Abelian	Real line		0	1
3	S3 = Sp(1) = SU(2)=Spin(3), SO3(${\displaystyle \mathbb {R} }$) = PSU(2)	Compact
3	SL2(${\displaystyle \mathbb {R} }$) = Sp2(${\displaystyle \mathbb {R} }$), SO2,1(${\displaystyle \mathbb {R} }$)	Split, Hermitian, hyperbolic	Hyperbolic plane ${\displaystyle \mathbb {H} ^{2}}$	Sphere S2	1	2
6	SL2(${\displaystyle \mathbb {C} }$) = Sp2(${\displaystyle \mathbb {C} }$), SO3,1(${\displaystyle \mathbb {R} }$), SO3(${\displaystyle \mathbb {C} }$)	Complex	Hyperbolic space ${\displaystyle \mathbb {H} ^{3}}$	Sphere S3	1	3
8	SL3(${\displaystyle \mathbb {R} }$)	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(${\displaystyle \mathbb {R} }$)	Compact
10	SO4,1(${\displaystyle \mathbb {R} }$), Sp2,2(${\displaystyle \mathbb {R} }$)	Hyperbolic, quaternionic	Hyperbolic space ${\displaystyle \mathbb {H} ^{4}}$	Sphere S4	1	4
10	SO3,2(${\displaystyle \mathbb {R} }$), Sp4(${\displaystyle \mathbb {R} }$)	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(${\displaystyle \mathbb {R} }$)	Compact
15	SL4(${\displaystyle \mathbb {R} }$), SO3,3(${\displaystyle \mathbb {R} }$)	Split	${\displaystyle \mathbb {R} }$3 in ${\displaystyle \mathbb {R} }$3,3	Grassmannian G(3,3)	3	9
15	SU(3,1)	Hermitian	Complex hyperbolic space	Complex projective space	1	6
15	SU(2,2), SO4,2(${\displaystyle \mathbb {R} }$)	Hermitian, quasi-split, quaternionic	${\displaystyle \mathbb {R} }$2 in ${\displaystyle \mathbb {R} }$2,4	Grassmannian G(2,4)	2	8
15	SL2(${\displaystyle \mathbb {H} }$), SO5,1(${\displaystyle \mathbb {R} }$)	Hyperbolic	Hyperbolic space ${\displaystyle \mathbb {H} ^{5}}$	Sphere S5	1	5
16	SL3(${\displaystyle \mathbb {C} }$)	Complex		SU(3)	2	8
20	SO5(${\displaystyle \mathbb {C} }$), Sp4(${\displaystyle \mathbb {C} }$)	Complex		Spin5(${\displaystyle \mathbb {R} }$)	2	10
21	SO7(${\displaystyle \mathbb {R} }$)	Compact
21	SO6,1(${\displaystyle \mathbb {R} }$)	Hyperbolic	Hyperbolic space ${\displaystyle \mathbb {H} ^{6}}$	Sphere S6
21	SO5,2(${\displaystyle \mathbb {R} }$)	Hermitian
21	SO4,3(${\displaystyle \mathbb {R} }$)	Split, quaternionic
21	Sp(3)	Compact
21	Sp6(${\displaystyle \mathbb {R} }$)	Split, hermitian
21	Sp4,2(${\displaystyle \mathbb {R} }$)	Quaternionic
24	SU(5)	Compact
24	SL5(${\displaystyle \mathbb {R} }$)	Split
24	SU4,1	Hermitian
24	SU3,2	Hermitian, quaternionic
28	SO8(${\displaystyle \mathbb {R} }$)	Compact
28	SO7,1(${\displaystyle \mathbb {R} }$)	Hyperbolic	Hyperbolic space ${\displaystyle \mathbb {H} ^{7}}$	Sphere S7
28	SO6,2(${\displaystyle \mathbb {R} }$)	Hermitian
28	SO5,3(${\displaystyle \mathbb {R} }$)	Quasi-split
28	SO4,4(${\displaystyle \mathbb {R} }$)	Split, quaternionic
28	SO∗8(${\displaystyle \mathbb {R} }$)	Hermitian
28	G2(${\displaystyle \mathbb {C} }$)	Complex
30	SL4(${\displaystyle \mathbb {C} }$)	Complex

Simply laced groups

A simply laced group is a Lie group whose Dynkin 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.

See also

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

References

• 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.

Further reading

• Besse, Einstein manifolds ISBN 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