Simple Lie group

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, $\mathbb {R}$ , 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 simple Lie groups were first classified by Wilhelm Killing and later perfected by Élie Cartan. This 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 $ℝ$ 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 $ℝ$ 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.

Alternatives

An equivalent definition of a simple Lie group follows from the Lie correspondence: A connected Lie group is simple if its Lie algebra is simple. An important technical point is that a simple Lie group may contain discrete normal subgroups, hence being a simple Lie group is different from being simple as an abstract group.

Simple Lie groups include many classical Lie groups, which provide a group-theoretic underpinning for spherical geometry, projective geometry and related geometries in the sense of Felix Klein's Erlangen program. It emerged in the course of classification of simple Lie groups that there exist also several exceptional possibilities not corresponding to any familiar geometry. These exceptional groups account for many special examples and configurations in other branches of mathematics, as well as contemporary theoretical physics.

As a counterexample, the general linear group is neither simple, nor semisimple. This is because multiples of the identity form a nontrivial normal subgroup, thus evading the definition. Equivalently, the corresponding Lie algebra has a degenerate Killing 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 the special orthogonal groups in even dimension. These have the matrix $-I$ in the center, and this element is path-connected to the identity element, and so these groups evade the definition. Both of these are reductive groups.

Related ideas

Simple Lie algebras

The Lie algebra of a simple Lie group is a simple Lie algebra. This is a one-to-one correspondence between connected simple Lie groups with trivial 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 their Dynkin diagrams, of types "ABCDEFG". If L is a real simple Lie algebra, its complexification is a simple complex Lie algebra, unless L is already the complexification of a Lie algebra, in which case the complexification of L is a product of two copies of L. This reduces the problem of classifying the real simple Lie algebras to that of finding all the real 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

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 each non-compact simple Lie group G, one compact and one non-compact. The non-compact one is a cover of the quotient of G by a maximal compact subgroup H, and the compact one is a cover of the quotient of the compact form of G by the same subgroup H. This duality between compact and non-compact symmetric spaces is a generalization of the well known duality between spherical and hyperbolic geometry.

Hermitian symmetric spaces

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 for p = 2, D III, and C I, and the two exceptional ones are types E III and E VII of complex dimensions 16 and 27.

Notation

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

In the symbols such as E₆⁻²⁶ 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).

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 ${\mathfrak {g}}$ , there is a unique "centerless" simple Lie group $G$ whose Lie algebra is ${\mathfrak {g}}$ 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 $K\subset \pi _{1}(G)$ of the fundamental group of some Lie group $G$ , one can use the theory of covering spaces to construct a new group ${\tilde {G}}^{K}$ with $K$ 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 $K\subset \pi _{1}(G)$ the full fundamental group, the resulting Lie group ${\tilde {G}}^{K=\pi _{1}(G)}$ is the universal cover of the centerless Lie group $G$ , and is simply connected. In particular, every (real or complex) Lie algebra also corresponds to a unique connected and simply connected Lie group ${\tilde {G}}$ with that Lie algebra, called the "simply connected Lie group" associated to ${\mathfrak {g}}.$

Compact Lie groups

Every simple complex Lie algebra has a unique real form whose corresponding centerless Lie group is compact. 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 the Peter–Weyl theorem. Just like simple complex Lie algebras, centerless compact Lie groups are classified by Dynkin diagrams (first classified by Wilhelm Killing and Élie Cartan).

For the infinite (A, B, C, D) series of Dynkin diagrams, the simply 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.

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.

List

Abelian

	Dimension	Outer automorphism group	Dimension of symmetric space	Symmetric space	Remarks
$ℝ$ (Abelian)	1	$ℝ$ ^∗	1	$ℝ$	^†

Notes

^† The group

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

ℝ

is the only such non-compact symmetric space without a compact dual (although it has a compact quotient S¹).

Compact

	Dimension	Fundamental group	Outer automorphism group	Other names	Remarks
A_n (n ≥ 1) compact	n(n + 2)	Cyclic, order n + 1	1 if n = 1, 2 if n > 1.	projective special unitary group PSU(n + 1)	A₁ is the same as B₁ and C₁
B_n (n ≥ 2) compact	n(2n + 1)	2	1	special orthogonal group SO_2n+1(R)	B₁ is the same as A₁ and C₁. B₂ is the same as C₂.
C_n (n ≥ 3) compact	n(2n + 1)	2	1	projective compact symplectic group PSp(n), PSp(2n), PUSp(n), PUSp(2n)	Hermitian. Complex structures of Hⁿ. Copies of complex projective space in quaternionic projective space.
D_n (n ≥ 4) compact	n(2n − 1)	Order 4 (cyclic when n is odd).	2 if n > 4, S₃ if n = 4	projective special orthogonal group PSO_2n(R)	D₃ is the same as A₃, D₂ is the same as A₁², and D₁ is abelian.
E₆⁻⁷⁸ compact	78	3	2
E₇⁻¹³³ compact	133	2	1
E₈⁻²⁴⁸ compact	248	1	1
F₄⁻⁵² compact	52	1	1
G₂⁻¹⁴ compact	14	1	1		This is the automorphism group of the Cayley algebra.

Split

	Dimension	Real rank	Maximal compact subgroup	Fundamental group	Outer automorphism group	Other names	Dimension of symmetric space	Compact symmetric space	Non-Compact symmetric space	Remarks
A_n I (n ≥ 1) split	n(n + 2)	n	D_n/2 or B_(n−1)/2	Infinite cyclic if n = 1 2 if n ≥ 2	1 if n = 1 2 if n ≥ 2.	projective special linear group PSL_n+1(R)	n(n + 3)/2	Real structures on Cⁿ⁺¹ or set of RPⁿ in CPⁿ. Hermitian if n = 1, in which case it is the 2-sphere.	Euclidean structures on Rⁿ⁺¹. Hermitian if n = 1, when it is the upper half plane or unit complex disc.
B_n I (n ≥ 2) split	n(2n + 1)	n	SO(n)SO(n+1)	Non-cyclic, order 4	1	identity component of special orthogonal group SO(n,n+1)	n(n + 1)			B₁ is the same as A₁.
C_n I (n ≥ 3) split	n(2n + 1)	n	A_n−1S¹	Infinite cyclic	1	projective symplectic group PSp_2n(R), PSp(2n,R), PSp(2n), PSp(n,R), PSp(n)	n(n + 1)	Hermitian. Complex structures of Hⁿ. Copies of complex projective space in quaternionic projective space.	Hermitian. Complex structures on R²ⁿ compatible with a symplectic form. Set of complex hyperbolic spaces in quaternionic hyperbolic space. Siegel upper half space.	C₂ is the same as B₂, and C₁ is the same as B₁ and A₁.
D_n I (n ≥ 4) split	n(2n - 1)	n	SO(n)SO(n)	Order 4 if n odd, 8 if n even	2 if n > 4, S₃ if n = 4	identity component of projective special orthogonal group PSO(n,n)	n²			D₃ is the same as A₃, D₂ is the same as A₁², and D₁ is abelian.
E₆⁶ I split	78	6	C₄	Order 2	Order 2	E I	42
E₇⁷ V split	133	7	A₇	Cyclic, order 4	Order 2		70
E₈⁸ VIII split	248	8	D₈	2	1	E VIII	128			@ E8
F₄⁴ I split	52	4	C₃ × A₁	Order 2	1	F I	28	Quaternionic projective planes in Cayley projective plane.	Hyperbolic quaternionic projective planes in hyperbolic Cayley projective plane.
G₂² I split	14	2	A₁ × A₁	Order 2	1	G I	8	Quaternionic subalgebras of the Cayley algebra. Quaternion-Kähler.	Non-division quaternionic subalgebras of the non-division Cayley algebra. Quaternion-Kähler.

Complex

	Real dimension	Real rank	Maximal compact subgroup	Fundamental group	Outer automorphism group	Other names	Dimension of symmetric space	Compact symmetric space	Non-Compact symmetric space
A_n (n ≥ 1) complex	2n(n + 2)	n	A_n	Cyclic, order n + 1	2 if n = 1, 4 (noncyclic) if n ≥ 2.	projective complex special linear group PSL_n+1(C)	n(n + 2)	Compact group A_n	Hermitian forms on Cⁿ⁺¹ with fixed volume.
B_n (n ≥ 2) complex	2n(2n + 1)	n	B_n	2	Order 2 (complex conjugation)	complex special orthogonal group SO_2n+1(C)	n(2n + 1)	Compact group B_n
C_n (n ≥ 3) complex	2n(2n + 1)	n	C_n	2	Order 2 (complex conjugation)	projective complex symplectic group PSp_2n(C)	n(2n + 1)	Compact group C_n
D_n (n ≥ 4) complex	2n(2n − 1)	n	D_n	Order 4 (cyclic when n is odd)	Noncyclic of order 4 for n > 4, or the product of a group of order 2 and the symmetric group S₃ when n = 4.	projective complex special orthogonal group PSO_2n(C)	n(2n − 1)	Compact group D_n
E₆ complex	156	6	E₆	3	Order 4 (non-cyclic)		78	Compact group E₆
E₇ complex	266	7	E₇	2	Order 2 (complex conjugation)		133	Compact group E₇
E₈ complex	496	8	E₈	1	Order 2 (complex conjugation)		248	Compact group E₈
F₄ complex	104	4	F₄	1	2		52	Compact group F₄
G₂ complex	28	2	G₂	1	Order 2 (complex conjugation)		14	Compact group G₂

Others

	Dimension	Real rank	Maximal compact subgroup	Fundamental group	Outer automorphism group	Other names	Dimension of symmetric space	Compact symmetric space	Non-Compact symmetric space	Remarks
A_2n−1 II (n ≥ 2)	(2n − 1)(2n + 1)	n − 1	C_n	Order 2		SL_n(H), SU^∗(2n)	(n − 1)(2n + 1)	Quaternionic structures on C²ⁿ compatible with the Hermitian structure	Copies of quaternionic hyperbolic space (of dimension n − 1) in complex hyperbolic space (of dimension 2n − 1).
A_n III (n ≥ 1) p + q = n + 1 (1 ≤ p ≤ q)	n(n + 2)	p	A_p−1A_q−1S¹			SU(p,q), A III	2pq	Hermitian. Grassmannian of p subspaces of C^p+q. If p or q is 2; quaternion-Kähler	Hermitian. Grassmannian of maximal positive definite subspaces of C^p,q. If p or q is 2, quaternion-Kähler	If p=q=1, split If \|p−q\| ≤ 1, quasi-split
B_n I (n > 1) p+q = 2n+1	n(2n + 1)	min(p,q)	SO(p)SO(q)			SO(p,q)	pq	Grassmannian of R^ps in R^p+q. If p or q is 1, Projective space If p or q is 2; Hermitian If p or q is 4, quaternion-Kähler	Grassmannian of positive definite R^ps in R^p,q. If p or q is 1, Hyperbolic space If p or q is 2, Hermitian If p or q is 4, quaternion-Kähler	If \|p−q\| ≤ 1, split.
C_n II (n > 2) n = p+q (1 ≤ p ≤ q)	n(2n + 1)	min(p,q)	C_pC_q	Order 2	1 if p ≠ q, 2 if p = q.	Sp_2p,2q(R)	4pq	Grassmannian of H^ps in H^p+q. If p or q is 1, quaternionic projective space in which case it is quaternion-Kähler.	H^ps in H^p,q. If p or q is 1, quaternionic hyperbolic space in which case it is quaternion-Kähler.
D_n I (n ≥ 4) p+q = 2n	n(2n − 1)	min(p,q)	SO(p)SO(q)		If p and q ≥ 3, order 8.	SO(p,q)	pq	Grassmannian of R^ps in R^p+q. If p or q is 1, Projective space If p or q is 2 ; Hermitian If p or q is 4, quaternion-Kähler	Grassmannian of positive definite R^ps in R^p,q. If p or q is 1, Hyperbolic Space If p or q is 2, Hermitian If p or q is 4, quaternion-Kähler	If p = q, split If \|p−q\| ≤ 2, quasi-split
D_n III (n ≥ 4)	n(2n − 1)	⌊n/2⌋	A_n−1R¹	Infinite cyclic	Order 2	SO^*(2n)	n(n − 1)	Hermitian. Complex structures on R²ⁿ compatible with the Euclidean structure.	Hermitian. Quaternionic quadratic forms on R²ⁿ.
E₆² II (quasi-split)	78	4	A₅A₁	Cyclic, order 6	Order 2	E II	40	Quaternion-Kähler.	Quaternion-Kähler.	Quasi-split but not split.
E₆⁻¹⁴ III	78	2	D₅S¹	Infinite cyclic	Trivial	E III	32	Hermitian. Rosenfeld elliptic projective plane over the complexified Cayley numbers.	Hermitian. Rosenfeld hyperbolic projective plane over the complexified Cayley numbers.
E₆⁻²⁶ IV	78	2	F₄	Trivial	Order 2	E IV	26	Set of Cayley projective planes in the projective plane over the complexified Cayley numbers.	Set of Cayley hyperbolic planes in the hyperbolic plane over the complexified Cayley numbers.
E₇⁻⁵ VI	133	4	D₆A₁	Non-cyclic, order 4	Trivial	E VI	64	Quaternion-Kähler.	Quaternion-Kähler.
E₇⁻²⁵ VII	133	3	E₆S¹	Infinite cyclic	Order 2	E VII	54	Hermitian.	Hermitian.
E₈⁻²⁴ IX	248	4	E₇ × A₁	Order 2	1	E IX	112	Quaternion-Kähler.	Quaternion-Kähler.
F₄⁻²⁰ II	52	1	B₄ (Spin₉(R))	Order 2	1	F II	16	Cayley projective plane. Quaternion-Kähler.	Hyperbolic Cayley projective plane. Quaternion-Kähler.

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	$ℝ$ , 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( $ℝ$ ), SO_2,1( $ℝ$ )	Split, Hermitian, hyperbolic	Hyperbolic plane $ℍ$ ²	Sphere S²	1	2
6	SL₂( $ℂ$ ) = Sp₂( $ℂ$ ), SO_3,1( $ℝ$ ), SO₃( $ℂ$ )	Complex	Hyperbolic space $ℍ$ ³	Sphere S³	1	3
8	SL₃( $ℝ$ )	Split	Euclidean structures on $ℝ$ ³	Real structures on $ℂ$ 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	SO_4,1( $ℝ$ ), Sp_2,2( $ℝ$ )	Hyperbolic, quaternionic	Hyperbolic space $ℍ$ ⁴	Sphere S⁴	1	4
10	SO_3,2( $ℝ$ ), Sp₄( $ℝ$ )	Split, Hermitian	Siegel upper half space	Complex structures on $ℍ$ 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( $ℝ$ ), SO_3,3( $ℝ$ )	Split	$ℝ$ ³ in $ℝ$ ^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), SO_4,2( $ℝ$ )	Hermitian, quasi-split, quaternionic	$ℝ$ ² in $ℝ$ ^2,4	Grassmannian G(2,4)	2	8
15	SL₂( $ℍ$ ), SO_5,1( $ℝ$ )	Hyperbolic	Hyperbolic space $ℍ$ ⁵	Sphere S⁵	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	SO_6,1( $ℝ$ )	Hyperbolic	Hyperbolic space $ℍ$ ⁶	Sphere S⁶
21	SO_5,2( $ℝ$ )	Hermitian
21	SO_4,3( $ℝ$ )	Split, quaternionic
21	Sp(3)	Compact
21	Sp₆( $ℝ$ )	Split, hermitian
21	Sp_4,2( $ℝ$ )	Quaternionic
24	SU(5)	Compact
24	SL₅( $ℝ$ )	Split
24	SU_4,1	Hermitian
24	SU_3,2	Hermitian, quaternionic
28	SO₈( $ℝ$ )	Compact
28	SO_7,1( $ℝ$ )	Hyperbolic	Hyperbolic space $ℍ$ ⁷	Sphere S⁷
28	SO_6,2( $ℝ$ )	Hermitian
28	SO_5,3( $ℝ$ )	Quasi-split
28	SO_4,4( $ℝ$ )	Split, quaternionic
28	SO^∗₈( $ℝ$ )	Hermitian
28	G₂( $ℂ$ )	Complex
30	SL₄( $ℂ$ )	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.

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.