Compact group
Basic examples of compact Lie groups include[1] The classification theorem of compact Lie groups states that up to finite extensions and finite covers this exhausts the list of examples (which already includes some redundancies).
The restrictions on n are to avoid special isomorphisms among the various families for small values of n. For each of these groups, the center is known explicitly.
The classification is through the associated root system (for a fixed maximal torus), which in turn are classified by their Dynkin diagrams.
A key idea in the study of a connected compact Lie group K is the concept of a maximal torus, that is a subgroup T of K that is isomorphic to a product of several copies of
The maximal torus in a compact group plays a role analogous to that of the Cartan subalgebra in a complex semisimple Lie algebra.
has been chosen, one can define a root system and a Weyl group similar to what one has for semisimple Lie algebras.
The second approach uses the root system and applies to all connected compact Lie groups.
consists of nth roots of unity times the identity, a cyclic group of order
In general, the center can be expressed in terms of the root lattice and the kernel of the exponential map for the maximal torus.
[5] The general method shows, for example, that the simply connected compact group corresponding to the exceptional root system
Compact groups all carry a Haar measure,[6] which will be invariant by both left and right translation (the modulus function must be a continuous homomorphism to positive reals (R+, ×), and so 1).
Therefore, integrals are often computable quite directly, a fact applied constantly in number theory.
as an orthogonal direct sum of finite-dimensional subspaces of matrix entries for the irreducible representations of
Cartan's theorem states that Im(ρ) must itself be a Lie subgroup in the unitary group.
Throughout this section, we fix a connected compact Lie group K and a maximal torus T in K. Since T is commutative, Schur's lemma tells us that each irreducible representation
We now briefly describe the structures needed to formulate the theorem; more details can be found in the article on weights in representation theory.
Finally, we say that one weight is higher than another if their difference can be expressed as a linear combination of elements of
The result says that: The theorem of the highest weight for representations of K is then almost the same as for semisimple Lie algebras, with one notable exception: The concept of an integral element is different.
is determined by its restriction to T. The study of characters is an important part of the representation theory of compact groups.
Specifically, in Weyl's analysis of the representations of K, the hardest part of the theorem—showing that every dominant, analytically integral element is actually the highest weight of some representation—is proved in a totally different way from the usual Lie algebra construction using Verma modules.
[15] Ultimately, the irreducible representations of K are realized inside the space of continuous functions on K. We now consider the case of the compact group SU(2).
We take the maximal torus to be the set of matrices of the form According to the example discussed above in the section on representations of T, the analytically integral elements are labeled by integers, so that the dominant, analytically integral elements are non-negative integers
Thus, we recover much of the information about the representations that is usually obtained from the Lie algebra computation.
We now outline the proof of the theorem of the highest weight, following the original argument of Hermann Weyl.
We focus on the most difficult part of the theorem, showing that every dominant, analytically integral element is the highest weight of some (finite-dimensional) irreducible representation.
Weyl's proof of the character formula is analytic in nature and hinges on the fact that the
Specifically, if there were any additional terms in the numerator, the Weyl integral formula would force the norm of the character to be greater than 1.
is the highest weight of a representation; nevertheless, the expressions on the right-hand side of the character formula gives a well-defined set of functions
And by the Peter–Weyl theorem, the characters of the irreducible representations form an orthonormal basis for the space of square integrable class functions.
Inside a general semisimple Lie group there is a maximal compact subgroup, and the representation theory of such groups, developed largely by Harish-Chandra, uses intensively the restriction of a representation to such a subgroup, and also the model of Weyl's character theory.
