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Movie Title Year Distributor Notes Rev Formats Bedside Highway 1972 Palladium NonSex Der må være en sengekant! 1975 Palladium NonSex In mathematics, a compact (topological) group is a topological group whose topology is compact. Compact groups are a natural generalization of finite groups with the discrete topology and have properties that carry over in significant fashion. Compact groups have a well-understood theory, in relation to group actions and representation theory. In the following we will assume all groups are Hausdorff spaces. Contents 1 Compact Lie groups 1.1 Classification 1.2 Maximal tori and root systems 1.3 Fundamental group and center 2 Further examples 3 Haar measure 4 Representation theory 5 Representation theory of a connected compact Lie group 5.1 Representation theory of T 5.2 Representation theory of K 5.3 The Weyl character formula 5.4 The SU(2) case 5.5 An outline of the proof
6 Duality 7 From compact to non-compact groups 8 See also 9 References 10 Bibliography Compact Lie groups Lie groups form a class of topological groups, and the compact Lie groups have a particularly well-developed theory. Basic examples of compact Lie groups include[1] the circle group T and the torus groups Tn, the orthogonal groups O(n), the special orthogonal group SO(n) and its covering spin group Spin(n), the unitary group U(n) and the special unitary group SU(n), the symplectic group Sp(n), the compact forms of the exceptional Lie groups: G2, F4, E6, E7, and E8, 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). This classification is described in more detail in the next subsection.

Classification Given any compact Lie group G one can take its identity component G0, which is connected. The quotient group G/G0 is the group of components p0(G) which must be finite since G is compact. We therefore have a finite extension {\displaystyle 1\to G_{0}\to G\to \pi _{0}(G)\to 1.\,}1\to G_{0}\to G\to \pi _{0}(G)\to 1.\, Meanwhile, for connected compact Lie groups, we have the following result:[2] Theorem: Every connected compact Lie group is the quotient by a finite central subgroup of a product of a simply connected compact Lie group and a torus. Thus, the classification of connected compact Lie groups can in principle be reduced to knowledge of the simply connected compact Lie groups together with information about their centers. (For information about the center, see the section below on fundamental group and center.) Finally, every compact, connected, simply-connected Lie group K is a product of compact, connected, simply-connected simple Lie groups Ki each of which is isomorphic to exactly one of the following: The compact symplectic group {\displaystyle \operatorname {Sp} (n),\,n\geq 1}{\displaystyle \operatorname {Sp} (n),\,n\geq 1} The special unitary group {\displaystyle \operatorname {SU} (n),\,n\geq 3}{\displaystyle \operatorname {SU} (n),\,n\geq 3} The spin group {\displaystyle \operatorname {Spin} (n),\,n\geq 7}{\displaystyle \operatorname {Spin} (n),\,n\geq 7} or one of the five exceptional groups G2, F4, E6, E7, and E8. 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. The classification of compact, simply connected Lie groups is the same as the classification of complex semisimple Lie algebras. Indeed, if K is a simply connected compact Lie group, then the complexification of the Lie algebra of K is semisimple. Conversely, every complex semisimple Lie algebra has a compact real form isomorphic to the Lie algebra of a compact, simply connected Lie group. Maximal tori and root systems See also: Maximal torus and Root system 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 several copies of {\displaystyle S^{1}}S^{1} and that is not contained in any larger subgroup of this type. A basic example is the case {\displaystyle K=SU(n)}{\displaystyle K=SU(n)}, in which case we may take {\displaystyle T}T to be the group of diagonal elements in {\displaystyle K}K. A basic result is the torus theorem which states that every element of {\displaystyle K}K belongs to a maximal torus and that all maximal tori are conjugate. The maximal torus in a compact group plays a role analogous to that of the Cartan subalgebra in a complex semisimple Lie algebra. In particular, once a maximal torus {\displaystyle T\subset K}{\displaystyle T\subset K} has been chosen, one can define a root system and a Weyl group similar to what one has for semisimple Lie algebras.[3] These structures then play an essential role both in the classification of connected compact groups (described above) and in the representation theory of a fixed such group (described below). The root systems associated to the simple compact groups appearing in the classification of simply connected compact groups are as follows:[4] The special unitary groups {\displaystyle \operatorname {SU} (n)}\operatorname {SU} (n) correspond to the root system {\displaystyle A_{n-1}}{\displaystyle A_{n-1}} The odd spin groups {\displaystyle \operatorname {Spin} (2n+1)}{\displaystyle \operatorname {Spin} (2n+1)} correspond to the root system {\displaystyle B_{n}}B_{{n}} The compact symplectic groups {\displaystyle \operatorname {Sp} (n)}{\displaystyle \operatorname {Sp} (n)} correspond to the root system {\displaystyle C_{n}}C_{n} The even spin groups {\displaystyle \operatorname {Spin} (2n)}{\displaystyle \operatorname {Spin} (2n)} correspond to the root system {\displaystyle D_{n}}D_{n} The exceptional compact Lie groups correspond to the five exceptional root systems G2, F4, E6, E7, or E8 Fundamental group and center See also: Fundamental group § Lie groups It is important to know whether a connected compact Lie group is simply connected, and if not, to determine its fundamental group. For compact Lie groups, there are two basic approaches to computing the fundamental group. The first approach applies to the classical compact groups {\displaystyle \operatorname {SU} (n)}\operatorname {SU} (n), {\displaystyle \operatorname {U} (n)}\operatorname {U} (n), {\displaystyle \operatorname {SO} (n)}{\displaystyle \operatorname {SO} (n)}, and {\displaystyle \operatorname {Sp} (n)}{\displaystyle \operatorname {Sp} (n)} and proceeds by induction on {\displaystyle n}n. The second approach uses the root system and applies to all connected compact Lie groups. It is also important to know the center of a connected compact Lie group. The center of a classical group {\displaystyle G}G can easily be computed "by hand," and in most cases consists simply of whatever multiples of the identity are in {\displaystyle G}G. (The group SO(2) is an exception—the center is the whole group, even though most elements are not multiples of the identity.) Thus, for example, the center of {\displaystyle \operatorname {SU} (n)}\operatorname {SU} (n) consists of nth roots of unity times the identity, a cyclic group of order {\displaystyle n}n. 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 {\displaystyle G_{2}}G_{2} has trivial center. Thus, the compact {\displaystyle G_{2}}G_{2} group is one of very few simple compact groups that are simultaneously simply connected and center free. (The others are {\displaystyle F_{4}}F_{4} and {\displaystyle E_{8}}E_{8}.) Further examples Amongst groups that are not Lie groups, and so do not carry the structure of a manifold, examples are the additive group Zp of p-adic integers, and constructions from it. In fact any profinite group is a compact group. This means that Galois groups are compact groups, a basic fact for the theory of algebraic extensions in the case of infinite degree. Pontryagin duality provides a large supply of examples of compact commutative groups. These are in duality with abelian discrete groups. Haar measure See also: Peter–Weyl theorem 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). In other words, these groups are unimodular. Haar measure is easily normalized to be a probability measure, analogous to d?/2p on the circle. Such a Haar measure is in many cases easy to compute; for example for orthogonal groups it was known to Adolf Hurwitz, and in the Lie group cases can always be given by an invariant differential form. In the profinite case there are many subgroups of finite index, and Haar measure of a coset will be the reciprocal of the index. Therefore, integrals are often computable quite directly, a fact applied constantly in number theory. If {\displaystyle K}K is a compact group and {\displaystyle m}m is the associated Haar measure, the Peter–Weyl theorem provides a decomposition of {\displaystyle L^{2}(K,dm)}{\displaystyle L^{2}(K,dm)} as an orthogonal direct sum of finite-dimensional subspaces of matrix entries for the irreducible representations of {\displaystyle K}K. Representation theory The representation theory of compact groups (not necessarily Lie groups and not necessarily connected) was founded by the Peter–Weyl theorem.[7] Hermann Weyl went on to give the detailed character theory of the compact connected Lie groups, based on maximal torus theory.[8] The resulting Weyl character formula was one of the influential results of twentieth century mathematics. The combination of the Peter–Weyl theorem and the Weyl character formula led Weyl to a complete classification of the representations of a connected compact Lie group; this theory is described in the next section. A combination of Weyl's work and Cartan's theorem gives a survey of the whole representation theory of compact groups G . That is, by the Peter–Weyl theorem the irreducible unitary representations ? of G are into a unitary group (of finite dimension) and the image will be a closed subgroup of the unitary group by compactness. Cartan's theorem states that Im(?) must itself be a Lie subgroup in the unitary group. If G is not itself a Lie group, there must be a kernel to ?. Further one can form an inverse system, for the kernel of ? smaller and smaller, of finite-dimensional unitary representations, which identifies G as an inverse limit of compact Lie groups. Here the fact that in the limit a faithful representation of G is found is another consequence of the Peter–Weyl theorem. The unknown part of the representation theory of compact groups is thereby, roughly speaking, thrown back onto the complex representations of finite groups. This theory is rather rich in detail, but is qualitatively well understood. Representation theory of a connected compact Lie group Certain simple examples of the representation theory of compact Lie groups can be worked out by hand, such as the representations of the rotation group SO(3), the special unitary group SU(2), and the special unitary group SU(3). We focus here on the general theory. See also the parallel theory of representations of a semisimple Lie algebra. Throughout this section, we fix a connected compact Lie group K and a maximal torus T in K. Representation theory of T Since T is commutative, Schur's lemma tells us that each irreducible representation {\displaystyle \rho }\rho of T is one-dimensional: {\displaystyle \rho :T\rightarrow GL(1;\mathbb {C} )=\mathbb {C} ^{*}}{\displaystyle \rho :T\rightarrow GL(1;\mathbb {C} )=\mathbb {C} ^{*}}. Since, also, T is compact, {\displaystyle \rho }\rho must actually map into {\displaystyle S^{1}\subset \mathbb {C} }{\displaystyle S^{1}\subset \mathbb {C} }. To describe these representations concretely, we let {\displaystyle {\mathfrak {t}}}\mathfrak{t} be the Lie algebra of T and we write points {\displaystyle h\in T}{\displaystyle h\in T} as {\displaystyle h=e^{H},\quad H\in {\mathfrak {t}}}{\displaystyle h=e^{H},\quad H\in {\mathfrak {t}}}. In such coordinates, {\displaystyle \rho }\rho will have the form {\displaystyle \rho (e^{H})=e^{i\lambda (H)}}{\displaystyle \rho (e^{H})=e^{i\lambda (H)}} for some linear functional {\displaystyle \lambda }\lambda on {\displaystyle {\mathfrak {t}}}\mathfrak{t}. Now, since the exponential map {\displaystyle H\mapsto e^{H}}{\displaystyle H\mapsto e^{H}} is not injective, not every such linear functional {\displaystyle \lambda }\lambda gives rise to a well-defined map of T into {\displaystyle S^{1}}S^{1}. Rather, let {\displaystyle \Gamma }\Gamma denote the kernel of the exponential map: {\displaystyle \Gamma =\{H\in {\mathfrak {t}}|e^{2\pi H}=\operatorname {Id} \}}{\displaystyle \Gamma =\{H\in {\mathfrak {t}}|e^{2\pi H}=\operatorname {Id} \}}, where {\displaystyle \operatorname {Id} }{\displaystyle \operatorname {Id} } is the identity element of T. (We scale the exponential map here by a factor of {\displaystyle 2\pi }2\pi in order to avoid such factors elsewhere.) Then for {\displaystyle \lambda }\lambda to give a well-defined map {\displaystyle \rho }\rho , {\displaystyle \lambda }\lambda must satisfy {\displaystyle \lambda (H)\in \mathbb {Z} ,\quad H\in \Gamma }{\displaystyle \lambda (H)\in \mathbb {Z} ,\quad H\in \Gamma }, where {\displaystyle \mathbb {Z} }\mathbb {Z} is the set of integers.[9] A linear functional {\displaystyle \lambda }\lambda 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.[10] Suppose, for example, T is just the group {\displaystyle S^{1}}S^{1} of complex numbers {\displaystyle e^{i\theta }}e^{i\theta } of absolute value 1. The Lie algebra is the set of purely imaginary numbers, {\displaystyle H=i\theta ,\,\theta \in \mathbb {R} ,}{\displaystyle H=i\theta ,\,\theta \in \mathbb {R} ,} and the kernel of the (scaled) exponential map is the set of numbers of the form {\displaystyle in}in where {\displaystyle n}n is an integer. A linear functional {\displaystyle \lambda }\lambda takes integer values on all such numbers if and only if it is of the form {\displaystyle \lambda (i\theta )=k\theta }{\displaystyle \lambda (i\theta )=k\theta } for some integer {\displaystyle k}k. The irreducible representations of T in this case are one-dimensional and of the form {\displaystyle \rho (e^{i\theta })=e^{ik\theta },\quad k\in \mathbb {Z} }{\displaystyle \rho (e^{i\theta })=e^{ik\theta },\quad k\in \mathbb {Z} }. Representation theory of K See also: Lie algebra representation § Classifying finite-dimensional representations of Lie algebras Example of the weights of a representation of the group SU(3) The "eightfold way" representation of SU(3), as used in particle physics Black dots indicate the dominant integral elements for the group SU(3) We now let {\displaystyle \Sigma }\Sigma denote a finite-dimensional irreducible representation of K (over {\displaystyle \mathbb {C} }\mathbb {C} ). We then consider the restriction of {\displaystyle \Sigma }\Sigma to T. This restriction is not irreducible unless {\displaystyle \Sigma }\Sigma 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 {\displaystyle \lambda }\lambda as in the preceding subsection. If a given {\displaystyle \lambda }\lambda occurs at least once in the decomposition of the restriction of {\displaystyle \Sigma }\Sigma to T, we call {\displaystyle \lambda }\lambda a weight of {\displaystyle \Sigma }\Sigma . 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 {\displaystyle R\subset {\mathfrak {t}}}{\displaystyle R\subset {\mathfrak {t}}} is very similar to the construction for complex semisimple Lie algebras. Specifically, the weights are the nonzero weights for the adjoint action of T on the complexified Lie algebra of K. The root system R has all the usual properties of a root system, except that the elements of R may not span {\displaystyle {\mathfrak {t}}}\mathfrak{t}.[11] We then choose a base {\displaystyle \Delta }\Delta for R and we say that an integral element {\displaystyle \lambda }\lambda is dominant if {\displaystyle \lambda (\alpha )\geq 0}{\displaystyle \lambda (\alpha )\geq 0} for all {\displaystyle \alpha \in \Delta }{\displaystyle \alpha \in \Delta }. Finally, we say that one weight is higher than another if their difference can be expressed as a linear combination of elements of {\displaystyle \Delta }\Delta with non-negative coefficients. The irreducible finite-dimensional representations of K are then classified by a theorem of the highest weight,[12] which is closely related to the analogous theorem classifying representations of a semisimple Lie algebra. The result says that: (1) every irreducible representation has highest weight, (2) the highest weight is always a dominant, analytically integral element, (3) two irreducible representations with the same highest weight are isomorphic, and (4) every dominant, analytically integral element arises as the highest weight of an irreducible representation. 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. The weights {\displaystyle \lambda }\lambda of a representation {\displaystyle \Sigma }\Sigma are analytically integral in the sense described in the previous subsection

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