Simple Lie group

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.

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. For this reason, the definition of a simple Lie group is not equivalent to the definition of a Lie group that is 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 ${\displaystyle -I}$  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.

A semisimple Lie group is a connected Lie group so that its only closed connected abelian normal 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 a central product of simple Lie groups. The semisimple Lie groups are exactly the Lie groups whose Lie algebras are semisimple 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 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.

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.

${\displaystyle \mathbb {R,C,H,O} }$   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).

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 ${\displaystyle {\mathfrak {g}}.}$ 

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, 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 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.

	Dimension	Outer automorphism group	Dimension of symmetric space	Symmetric space	Remarks
${\displaystyle \mathbb {R} }$  (Abelian)	1	${\displaystyle \mathbb {R} ^{*}}$	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 quotient S¹).

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 ${\displaystyle \mathbb {H} ^{2}}$	Sphere S2	1	2
6	SL2(ℂ) = Sp2(ℂ), SO3,1(ℝ), SO3(ℂ)	Complex	Hyperbolic space ${\displaystyle \mathbb {H} ^{3}}$	Sphere S3	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 ${\displaystyle \mathbb {H} ^{4}}$	Sphere S4	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	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(ℝ)	Hermitian, quasi-split, quaternionic	ℝ2 in ℝ2,4	Grassmannian G(2,4)	2	8
15	SL2(ℍ), SO5,1(ℝ)	Hyperbolic	Hyperbolic space ${\displaystyle \mathbb {H} ^{5}}$	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	SO6,1(ℝ)	Hyperbolic	Hyperbolic space ${\displaystyle \mathbb {H} ^{6}}$	Sphere S6
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 ${\displaystyle \mathbb {H} ^{7}}$	Sphere S7
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

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.

• 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 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