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.

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.

An equivalent definition of a simple Lie group follows from the Lie correspondence: a connected Lie group is simple if its Lie algebra is a 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.

Classification of simple Lie groups

Main article: List of simple Lie groups

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 ${\displaystyle {\mathfrak {g}}.}$

Compact Lie groups

Main article: root system

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.

A series

A₁, A₂, ...

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 series

B₂, B₃, ...

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 series

C₃, C₄, ...

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.

D series

D₄, D₅, ...

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).

Exceptional cases

Main article: exceptional Lie algebra

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_7½.

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

References

• Jacobson, Nathan (1971). Exceptional Lie Algebras (1 ed.). CRC Press. ISBN 0-8247-1326-5.

• Fulton, WIlliam and Harris, Joe (2004). Representation Theory: A First Course. Springer.doi:10.1007/978-1-4612-0979-9