Simple Lie group

Simple Lie groups are a class of Lie groups which play a role in Lie group theory similar to that of simple groups in the theory of discrete groups. Essentially, simple Lie groups are connected Lie groups which cannot be decomposed as an extension of smaller connected Lie groups, and which are not commutative. Together with the commutative Lie group of the real numbers, $\mathbb {R}$ and that of the unit complex numbers, U(1), 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 group SL(n) of n by n matrices with determinant equal to 1 is simple for all n > 1.

In group theory, a simple Lie group is a connected locally compact non-abelian Lie group G which does not have nontrivial connected normal subgroups.

A simple Lie algebra is a non-abelian Lie algebra whose only ideals are 0 and itself (or equivalently, a Lie algebra of dimension 2 or more, whose only ideals are 0 and itself).

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 programme. 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.

Simple complex Lie groups

All (locally compact, connected) Lie groups are smooth manifolds. Mathematicians often study complex Lie groups, which are Lie groups with a complex structure on the underlying manifold, which is required to be compatible with the group operations. A complex Lie group is called simple if it is connected as a topological space and its Lie algebra is simple as a complex Lie algebra. Note that the underlying Lie group may not be simple, although it will still be semisimple (see below).

Semisimple and reductive groups

It is often useful to study slightly more general classes of Lie groups than simple groups, namely semisimple or, more generally, reductive Lie groups. A connected Lie group is called semisimple if its Lie algebra is a semisimple lie algebra, i.e. a direct sum of simple Lie algebras. It is called reductive if its Lie algebra is a direct sum of simple and trivial (one-dimensional) Lie algebras. Reductive groups occur naturally as symmetries of a number of mathematical objects in algebra, geometry, and physics. For example, the group $GL_{n}(\mathbb {R} )$ of symmetries of an n-dimensional real vector space (equivalently, the group of invertible matrices) is reductive.

Representation Theory

A topological group homomorphism from a Lie group $G$ to a matrix group is called a representation of $G$ , and representations of simple Lie groups are the building blocks of the branch of mathematics called representation theory. Finite-dimensional representations of simple groups split into direct sums of irreducible representations, which are classified by vectors in the weight lattice satisfying certain properties.

Classification 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 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 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 both its associated simply connected and centerless compact groups the odd special orthogonal groups, SO(2r + 1).

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 simply connected compact group the even special orthogonal groups, SO(2r) and as its associated centerless compact group the projective special orthogonal group PSO(2 r) = SO(2 r)/{I, -I}.

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

In addition to the four families above, there are five so-called exceptional Dynkin diagrams G₂, F₄, E₆, E₇, and E₈. All of these also have associated simply connected and centerless compact groups, although these are not as easy to describe in terms of matrix groups as the infinite series A_i, B_i, C_i and D_i above.

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 (1971-06-01). Exceptional Lie Algebras (1 ed.). CRC Press. ISBN 0-8247-1326-5.

Fulton, WIlliam and Harris, Joe. Representation Theory, A First Course, Springer Readings in Mathematics