Editing Compact group (section)

===Representation theory of ''T''===
Since ''T'' is commutative, [[Schur's lemma]] tells us that each irreducible representation <math>\rho</math> of ''T'' is one-dimensional:
:<math>\rho:T\rightarrow GL(1;\mathbb{C})=\mathbb{C}^* .</math>
Since, also, ''T'' is compact, <math>\rho</math> must actually map into <math>S^1\subset\mathbb{C}</math>.

To describe these representations concretely, we let <math>\mathfrak{t}</math> be the Lie algebra of ''T'' and we write points <math>h\in T</math> as
:<math>h=e^{H},\quad H\in\mathfrak{t} .</math>
In such coordinates, <math>\rho</math> will have the form
:<math>\rho(e^{H})=e^{i \lambda(H)}</math>
for some linear functional <math>\lambda</math> on <math>\mathfrak{t}</math>.

Now, since the exponential map <math>H\mapsto e^{H}</math> is not injective, not every such linear functional <math>\lambda</math> gives rise to a well-defined map of ''T'' into <math>S^1</math>. Rather, let <math>\Gamma</math> denote the kernel of the exponential map:
:<math>\Gamma = \left\{ H\in\mathfrak{t} \mid e^{2\pi H}=\operatorname{Id} \right\},</math>
where <math>\operatorname{Id}</math> is the identity element of ''T''. (We scale the exponential map here by a factor of <math>2\pi</math> in order to avoid such factors elsewhere.) 
Then for <math>\lambda</math> to give a well-defined map <math>\rho</math>, <math>\lambda</math> must satisfy
:<math>\lambda(H)\in\mathbb{Z},\quad H\in\Gamma,</math>
where <math>\mathbb{Z}</math> is the set of integers.<ref>{{harvnb|Hall|2015}} Proposition 12.9</ref> A linear functional <math>\lambda</math> satisfying this condition is called an '''analytically integral element'''. This integrality condition is related to, but not identical to, the notion of [[Weight (representation theory)#Weights in the representation theory of semisimple Lie algebras|integral element]] in the setting of semisimple Lie algebras.<ref>{{harvnb|Hall|2015}} Section 12.2</ref>

Suppose, for example, ''T'' is just the group <math>S^1</math> of complex numbers <math>e^{i\theta}</math> of absolute value 1. The Lie algebra is the set of purely imaginary numbers, <math>H=i\theta,\,\theta\in\mathbb{R},</math> and the kernel of the (scaled) exponential map is the set of numbers of the form <math>in</math> where <math>n</math> is an integer. A linear functional <math>\lambda</math> takes integer values on all such numbers if and only if it is of the form <math>\lambda(i\theta)= k\theta</math> for some integer <math>k</math>. The irreducible representations of ''T'' in this case are one-dimensional and of the form
:<math>\rho(e^{i\theta})=e^{ik\theta},\quad k \in \Z .</math>