concretely, we let be the Lie algebra of T and we write points as In such coordinates, will have the form for some linear functional on . Now, since the exponential map is not injective, not every such linear functional gives rise to a well-defined map of T into . Rather, let denote the kernel of the exponential map: where is the identity element of T. (We scale the exponential map here by a factor of in order to avoid such factors elsewhere.) Then for to give a well-defined map , must satisfy where is the set of integers. A linear functional satisfying this condition is called an analytically integral element. This integrality condition is related to, but not identical to, the notion of integral element in the setting of semisimple Lie algebras. Suppose, for example, T is just the group of complex numbers of absolute value 1. The Lie algebra is the set of purely imaginary numbers, and the kernel of the (scaled) exponential map is the set of numbers of the form where is an integer. A linear functional takes integer values on all such numbers if and only if it is of the form for some integer . The irreducible representations of T in this case are one-dimensional and of the form Representation theory of K We now let denote a finite-dimensional irreducible representation of K (over ). We then consider the restriction of to T. This restriction is not irreducible unless is one-dimensional. Nevertheless, the restriction decomposes as a direct sum of irreducible representations of T. (Note that a given irreducible representation of T may occur more than once.) Now, each irreducible representation of T is described by a linear functional as in the preceding subsection. If a given occurs at least once in the decomposition of the restriction of to T, we call a weight of . The strategy of the representation theory of K is to classify the irreducible representations in terms of their weights. We now briefly describe the structures needed to formulate the theorem; more details can be found in the article on weights in representation theory. We need the notion of a root system for K (relative to a given maximal torus T). The construction of this root system is very similar to the construction for complex semisimple Lie algebras. Specifically, the weights are the nonzero weights for the adjoint action