Compact group
A compact group is a topological group G that is compact as a topological space, meaning the underlying topology is Hausdorff, the group multiplication and inversion maps are continuous, and every open cover of G has a finite subcover.[1] This structure combines algebraic group properties with the geometric constraint of compactness, ensuring that G is both totally bounded and complete in its metric realization.[2] Compact groups play a central role in harmonic analysis and representation theory due to their rich structural properties.[3] Every compact group admits a unique (up to positive scalar multiple) bi-invariant Haar measure, which is finite and turns G into a probability space when normalized.[1] Consequently, the space of square-integrable functions L^2(G) with respect to this measure decomposes orthogonally into finite-dimensional irreducible unitary representations via the Peter-Weyl theorem, which states that the matrix coefficients of these representations are dense in C(G), the continuous functions on G.[4] This theorem implies that all continuous unitary representations of compact groups are completely reducible, generalizing the decomposition of representations for finite groups.[5] Examples of compact groups abound in both abstract and concrete settings. The circle group \mathbb{T} = \mathbb{R}/\mathbb{Z}, equipped with addition modulo 1, is the archetypal abelian compact group.[6] Classical Lie groups such as the unitary group U(n) and the orthogonal group O(n) are compact matrix groups under matrix multiplication, arising naturally in linear algebra and geometry.[3] More generally, profinite groups like the p-adic integers \mathbb{Z}_p provide non-Lie examples, and any finite group with the discrete topology is compact.[7] For abelian compact groups, Pontryagin duality identifies them with discrete abelian groups, establishing a profound correspondence between compact and discrete structures.[8] These properties make compact groups indispensable in applications ranging from quantum mechanics, where symmetry groups like SU(2) model spin, to number theory and algebraic geometry via their connections to profinite completions.[3]Definition and properties
Definition
In mathematics, a compact group is a topological group G in which the underlying topological space is compact, meaning that every open cover of G admits a finite subcover.[9] Compact groups are typically assumed to be Hausdorff, ensuring that the topology separates points.[9] For a Hausdorff compact group, equivalent formulations arise in the metrizable case: the induced uniformity makes G a complete and totally bounded metric space.[9] More generally, arbitrary products of compact groups are compact by Tychonoff's theorem, facilitating the study of infinite-dimensional examples via inverse limits.[9] The term and concept of compact groups were introduced by Hermann Weyl in 1925, initially in the context of representation theory for compact Lie groups such as the special unitary group SU(n).[10] This framework was later generalized beyond Lie groups. A basic non-trivial example is the circle group \mathbb{T} = U(1), consisting of complex numbers of modulus 1 under multiplication, which is compact as a subset of \mathbb{C}.[9] Compactness ensures the existence of a bi-invariant Haar measure on G.[9]Basic topological and algebraic properties
In a compact topological group G, the inversion map g \mapsto g^{-1} is a continuous homeomorphism, and the multiplication map (g, h) \mapsto gh from G \times G to G is continuous and uniformly continuous with respect to the respective uniform structures on G and G \times G.[11] The continuity of these operations follows from the definition of a topological group, but compactness ensures additional regularity: specifically, the uniform continuity of multiplication arises because G \times G is compact and the map is continuous, implying that preimages of entourages (basic neighborhoods in the uniform structure) are open and thus contain compact sets whose finite covers yield uniform bounds.[11] A sketch of the proof for uniform continuity involves showing that for any entourage W in the uniformity of G, the preimage under multiplication is an open set in G \times G; since G \times G is compact, this preimage admits a finite cover by basic entourages, establishing the uniform property.[11] The conjugacy class of any element g \in G, defined as \{ h g h^{-1} \mid h \in G \}, is the continuous image of the compact space G under the map h \mapsto h g h^{-1}, and thus compact.[11] Since G is Hausdorff, this image is also closed, making each conjugacy class a compact closed subset of G.[11] If G is discrete, its compactness implies that G is finite, so every conjugacy class is finite.[11] Compact groups have no small subgroups, meaning there exists a neighborhood V of the identity e such that no nontrivial subgroup of G is contained in V.[11] This contrasts with the local structure of noncompact Lie groups, where neighborhoods of the identity approximate Lie algebra elements but do not form subgroups globally. The proof relies on compactness: if every neighborhood contained a nontrivial subgroup, repeated generation would yield a proper closed infinite subgroup whose compactness leads to a contradiction via finite index or covering arguments.[11] A key consequence of compactness is that for any neighborhood U of the identity e, the collection of left translates \{ g U \mid g \in G \} forms an open cover of G. By compactness, there exists a finite subcover, so G = \bigcup_{i=1}^n g_i U for some finite set \{ g_1, \dots, g_n \} \subset G.[11] This finite covering property underscores the "discreteness at infinity" in compact groups, where local neighborhoods suffice to cover the entire space finitely.[11]Examples
Abelian compact groups
Abelian compact groups form an important subclass of compact groups, characterized by their commutative operation, which simplifies their structural analysis through tools like Pontryagin duality. These groups arise naturally in harmonic analysis and topological group theory, where their duals provide insights into discrete structures.[12] A fundamental result is Pontryagin duality, which establishes that every compact abelian group G is topologically isomorphic to the Pontryagin dual of some discrete abelian group. Specifically, the Pontryagin dual \hat{G} of G consists of all continuous homomorphisms from G to the circle group \mathbb{T} = \mathbb{R}/\mathbb{Z}, equipped with the compact-open topology, and this dual is discrete. Conversely, the dual of a discrete abelian group is compact abelian. This duality interchanges compactness and discreteness, enabling a complete classification via the structure of discrete abelian groups, which decompose as direct sums of cyclic groups.[12] The algebraic and topological structure of compact abelian groups reflects this duality: every such group G decomposes as a topological direct product G \cong G_0 \times D, where G_0 is the connected component of the identity (a compact connected abelian group) and D is totally disconnected. The connected part G_0 is divisible in the case of tori. Examples of G_0 include finite-dimensional tori \mathbb{T}^n, which occur when G_0 is a Lie group, and more generally solenoid groups, which are non-Lie connected compact abelian groups.[13] The totally disconnected part D is profinite, meaning it is the inverse limit of an inverse system of finite abelian groups under continuous surjective homomorphisms. In general, compact abelian groups themselves can be viewed through this lens, with the theorem that they arise as inverse limits of finite abelian groups capturing the profinite component's role in the overall decomposition.[14][12] Representative examples illustrate this structure. The n-torus \mathbb{T}^n = (\mathbb{T})^n, for finite n, is a connected compact abelian Lie group, serving as the dual of the discrete group \mathbb{Z}^n; it exemplifies the connected divisible case and appears in applications like multidimensional Fourier analysis. Profinite completions provide totally disconnected examples, such as the p-adic integers \mathbb{Z}_p for a prime p, which is the inverse limit \varprojlim \mathbb{Z}/p^n\mathbb{Z} and the dual of the Prüfer p-group \mathbb{Z}(p^\infty); this group is torsion-free and metrizable. An uncountable torsion-free example is the additive group of the p-adic integers \mathbb{Z}_p, which is homeomorphic to the Cantor set and highlights the existence of non-Lie compact abelian structures beyond finite products.[14][12]Compact Lie groups
A compact Lie group is a Lie group endowed with a compact topology, where a Lie group is defined as a smooth manifold G equipped with a group structure such that the multiplication map m: G \times G \to G and the inversion map i: G \to G are smooth.[15][16] The compatibility between the manifold structure and the group operations ensures that the topology on G serves as both a manifold topology and a topological group topology, making every compact Lie group a compact topological group.[17] This compactness imposes strong restrictions on the group's structure, distinguishing compact Lie groups from non-compact ones like \mathrm{SL}(n, \mathbb{R}).[18] Prominent examples of compact Lie groups include the classical series: the special orthogonal groups \mathrm{SO}(n) consisting of n \times n real orthogonal matrices with determinant 1, the special unitary groups \mathrm{SU}(n) of n \times n complex unitary matrices with determinant 1, the unitary groups \mathrm{U}(n), and the compact symplectic groups \mathrm{Sp}(n) acting as quaternionic isometries on \mathbb{H}^n.[18][17] Additionally, there are five exceptional compact simple Lie groups: G_2, F_4, E_6, E_7, and E_8, which arise from unique root systems and have dimensions 14, 52, 78, 133, and 248, respectively.[18][17] These groups, along with their products and finite covers like the spin groups \mathrm{Spin}(n), illustrate the diversity within this class.[18] Compactness yields distinctive algebraic properties for these groups. On the Lie algebra \mathfrak{g}, the Killing form K(X, Y) = \operatorname{tr}(\operatorname{ad}_X \operatorname{ad}_Y) is negative definite for any nonzero X \in \mathfrak{g} when \mathfrak{g} is semisimple, providing an Ad-invariant inner product that endows \mathfrak{g} with a positive definite metric via -\langle X, Y \rangle = -K(X, Y).[19][20] Furthermore, all adjoint orbits under the action \operatorname{Ad}: G \to \mathrm{GL}(\mathfrak{g}) are closed, as the image of the compact group G under the continuous adjoint map is compact and hence closed in the Hausdorff topology of \mathfrak{g}.[18] These features facilitate the study of representations and structure, with classification relying on root systems associated to maximal tori.[17]Totally disconnected compact groups
A totally disconnected compact group is a compact topological group in which the connected component of the identity element is trivial, meaning that the only connected subgroups are the trivial one.[21] Equivalently, such a group admits a basis of neighborhoods of the identity consisting of open subgroups, ensuring that every neighborhood of the identity contains no nontrivial connected subsets.[22] The structure of totally disconnected compact groups is captured by their identification as profinite groups, which are inverse limits of finite discrete groups.[9] A fundamental theorem states that every totally disconnected compact group is profinite, possessing a basis of neighborhoods of the identity formed by open normal subgroups of finite index.[21] This profinite nature implies that these groups are Stone spaces in their dual formulation, with the topology arising from the inverse limit construction.[23] Representative examples include the profinite completion of the integers, denoted \hat{\mathbb{Z}}, which is the inverse limit \varprojlim_n \mathbb{Z}/n\mathbb{Z} and serves as the universal profinite quotient of \mathbb{Z}.[24] Another key example is the general linear group \mathrm{GL}_n(\mathbb{Z}_p) over the p-adic integers \mathbb{Z}_p for a prime p, which is compact and totally disconnected as a p-adic Lie group without a nontrivial connected component.[25] Additionally, closed automorphism groups of locally finite trees, such as certain rigid trees, yield compact totally disconnected subgroups when restricted to fixed-point-free actions preserving the tree structure.[21] These groups find significant applications in number theory, particularly through their role as absolute Galois groups, which are profinite and thus totally disconnected compact, governing the structure of algebraic extensions via the Krull topology.[23] For instance, the absolute Galois group of the rationals \mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q}) exemplifies how such structures encode infinite Galois theory, with fixed fields corresponding to open normal subgroups.[26] Haar measure on these groups exists and is normalized on compact open subgroups, facilitating integration over profinite completions in analytic number theory.[9]Haar measure
Existence and uniqueness
A Haar measure on a compact group G is defined as a regular Borel measure \mu on G that is left-invariant, meaning \mu(gA) = \mu(A) for all g \in G and Borel sets A \subseteq G, non-zero and finite on compact sets (with \mu(G) < \infty), and positive on non-empty open sets.[27][1] This measure induces a left-invariant integral on continuous functions f: G \to \mathbb{C}, satisfying \int_G f(g) \, d\mu(g) = \int_G f(hg) \, d\mu(g) for all h \in G and integrable f.[27] The fundamental theorem on Haar measure for compact groups states that there exists a unique (up to positive scalar multiple) left-invariant regular Borel measure \mu on G that is finite and positive on compact sets, and it can be normalized so that \mu(G) = 1, making it a probability measure.[27][1] For compact groups, this normalized Haar measure is also right-invariant, hence bi-invariant.[1] Existence follows from the Riesz representation theorem applied to the space of continuous functions C(G) on the compact group G, which is equipped with the sup norm. Since G is compact, C(G) separates points, and one constructs a positive linear functional \Lambda: C(G) \to \mathbb{C} that is left-invariant by approximating it via finite sums over group elements and using partitions of unity or mean values of translates. Specifically, for f \in C(G), define the mean value over a finite set \{a_1, \dots, a_n\} as \frac{1}{n} \sum_{i=1}^n f(ga_i), and take the limit in the uniform topology using compactness to obtain a translation-invariant functional, which represents a regular measure by Riesz.[27][1][28] Uniqueness up to scalar multiple is established by showing that if \mu and \nu are two left-invariant regular Borel measures on G, then there exists c > 0 such that \nu = c \mu. This relies on the fact that for any continuous f \geq 0 with \int f \, d\mu = 1, the translates f_h(g) = f(h^{-1}g) span a dense subspace, and invariance implies \int f_h \, d\nu = \int f \, d\nu for all h, so by density and continuity, \nu is a multiple of \mu. Normalization \mu(G) = 1 then fixes the constant.[27][1]Properties and normalization
One key property of the Haar measure on a compact group G is its bi-invariance: the left-invariant Haar measure \mu is also right-invariant, meaning \mu(Ag) = \mu(A) for all measurable A \subseteq G and g \in G.[1] This follows from the fact that compact groups are unimodular, so the modular function \Delta: G \to (0, \infty) satisfies \Delta(h) = 1 for all h \in G.[29] It is conventional to normalize the Haar measure on a compact group G such that \mu(G) = 1, making it a probability measure.[30] For a closed subgroup H \subseteq G, the quotient space G/H inherits a unique Haar measure \nu from \mu, defined via the disintegration formula \int_G f(g) \, d\mu(g) = \int_H \left( \int_{G/H} f(hx) \, d\nu(x) \right) d\mu_H(h) for suitable integrable f: G \to \mathbb{C}, where \mu_H is the normalized Haar measure on H.[31] This induced measure \nu is also normalized so that \nu(G/H) = 1, satisfying \mu(G) = \nu(G/H).[32] The finite total measure enables an analog of Fubini's theorem for products: on G \times G equipped with the product measure \mu \times \mu, integrals of measurable functions f: G \times G \to \mathbb{C} satisfy \int_{G \times G} f(g_1, g_2) \, d(\mu \times \mu)(g_1, g_2) = \int_G \left( \int_G f(g_1, g_2) \, d\mu(g_2) \right) d\mu(g_1) whenever the iterated integrals exist.[32] Bi-invariance implies a simple change-of-variables formula. For a right translate, the general relation for left Haar measures is \int_G f(g) \, d\mu(g) = \int_G f(gh) \Delta(h)^{-1} \, d\mu(g) for integrable f: G \to \mathbb{C} and h \in G.[29] In compact groups, \Delta \equiv 1, so this reduces to \int_G f(g) \, d\mu(g) = \int_G f(gh) \, d\mu(g), confirming right invariance directly.[1] The finiteness of \mu(G) ensures that the L^p spaces on G are well-behaved: for $1 \leq p < q \leq \infty, L^q(G) \subseteq L^p(G) with continuous inclusion, and the dual of L^p(G) is L^{p'}(G) where $1/p + 1/p' = 1.[30] This structure underpins harmonic analysis on compact groups, facilitating decompositions like the Peter-Weyl theorem.[31]Structure of compact groups
General structure theorem
The general structure theorem for compact groups, primarily due to the work of Andrew M. Gleason, Hidehiko Yamabe, Deane Montgomery, and Leo Zippin, characterizes their algebraic and topological form in terms of Lie and profinite components. Specifically, every connected compact Hausdorff group is a Lie group. This resolves the compact case of Hilbert's fifth problem, affirming that connectedness and compactness suffice for the group to admit a compatible Lie group structure, with smooth manifold topology and Lie algebra. More globally, every compact group arises as an extension of a compact Lie group by a profinite group: there exists a closed normal compact Lie subgroup L (the connected component of the identity) such that the quotient G/L is a totally disconnected compact group, hence profinite.[33] This structure implies that compact groups are pro-Lie groups, meaning they are inverse limits of Lie groups. To see this, given any neighborhood U of the identity in a compact Hausdorff group G, there exists a compact normal subgroup H \subseteq U such that G/H is a Lie group (in fact, linear over \mathbb{C}). Iterating over a basis of neighborhoods yields a system of surjective homomorphisms from G onto Lie groups with kernels forming a basis of neighborhoods, establishing the inverse limit description. Profinite groups themselves fit as the totally disconnected case, being inverse limits of finite discrete groups.[33] The proof outline leverages the no small subgroups (NSS) property: a topological group has NSS if there exists a neighborhood of the identity containing no nontrivial proper subgroup. Locally compact groups with NSS are precisely the Lie groups. For compact G, the Peter–Weyl theorem provides faithful finite-dimensional unitary representations, allowing linearization and approximation. One constructs open normal subgroups by quotienting out small kernels where the image inherits NSS (via metric approximations and continuity arguments), ensuring the quotients are Lie; the compactness ensures the kernels are compact and normal. This approximation process yields the Lie-by-profinite extension and the pro-Lie inverse limit.[33] A key corollary is that the dimension of a compact group G is well-defined as the dimension of its maximal connected Lie subgroup (the connected component of the identity), which coincides with the dimension of the associated Lie algebra. This dimension is invariant under the approximations and finite quotients in the structure theorem, providing a measure of the "Lie part" even for infinite-dimensional profinite extensions. For example, the additive group of p-adic integers \mathbb{Z}_p has dimension 0, as it is profinite with trivial connected component.[33]Structure of compact Lie groups
Compact Lie groups exhibit a canonical decomposition that separates their abelian and semisimple components. For a connected compact Lie group G with Lie algebra \mathfrak{g}, the Lie algebra decomposes as \mathfrak{g} = \mathfrak{z}(\mathfrak{g}) \oplus [\mathfrak{g}, \mathfrak{g}], where \mathfrak{z}(\mathfrak{g}) is the center of \mathfrak{g} (an abelian Lie algebra) and [\mathfrak{g}, \mathfrak{g}] is semisimple.[34] At the group level, G = Z(G)^0 \times [G, G], where Z(G)^0 denotes the connected component of the identity in the center Z(G) of G (a torus) and [G, G] is the commutator subgroup, a closed connected semisimple Lie subgroup with finite center.[35] In the semisimple case, where Z(G)^0 is trivial, the adjoint representation of G is faithful, with kernel precisely Z(G), which is finite.[35] Consequently, the adjoint form G / Z(G) has trivial center and acts faithfully via the adjoint representation; if the Lie algebra of G is simple, then G / Z(G) is a simple Lie group.[35] Simply connected compact Lie groups admit a product decomposition into a torus and a product of simple simply connected compact Lie groups, reflecting the direct sum structure of their semisimple Lie algebras.[36] This structure underscores the reductive nature of compact Lie algebras and facilitates the study of representations and homomorphisms.[35]Classification and geometry of compact Lie groups
Classification by rank and type
The classification of simple compact Lie algebras over the real numbers, which underpin the structure of simple compact Lie groups, divides them into four infinite families of classical types and five exceptional types, as established by the work of Killing and Cartan.[18] The classical families are A_n (corresponding to the special unitary Lie algebra su(n+1)), B_n (odd orthogonal so(2n+1)), C_n (compact symplectic sp(n)), and D_n (even orthogonal so(2n)), while the exceptional families are G_2, F_4, E_6, E_7, and E_8.[37][38] The rank of a simple compact Lie algebra is the dimension of its Cartan subalgebra, or equivalently, the dimension of the maximal torus in the corresponding Lie group. For the classical types A_n, B_n, C_n, and D_n, the rank is n (with n ≥ 1 for A_n, n ≥ 2 for B_n, n ≥ 3 for C_n to exclude isomorphisms C_1 ≅ A_1 and C_2 ≅ B_2, and n ≥ 4 for D_n to exclude D_2 ≅ A_1 × A_1 (not simple) and D_3 ≅ A_3). The exceptional types have fixed ranks: 2 for G_2, 4 for F_4, 6 for E_6, 7 for E_7, and 8 for E_8.[18][38] This distinction between classical (infinite families tied to matrix groups) and exceptional (finite, non-matrix-like) types highlights the organizational structure of the classification. Low-rank isomorphisms include A_1 ≅ B_1 ≅ C_1 and B_2 ≅ C_2.[37] Each simple complex semisimple Lie algebra admits a unique compact real form up to isomorphism, ensuring that the compact Lie groups associated with these algebras are determined uniquely by their underlying complex structure, modulo covering groups.[37][18] For instance, the Lie algebra su(2) of type A_1 is isomorphic to so(3) of type B_1, corresponding to the groups SU(2) and Spin(3), which are double covers of SO(3).[18] The following table summarizes the types, associated Lie algebras, ranks, dimensions (of the Lie algebra), and representative simply connected groups (noting low-rank isomorphisms: A_1 ≅ B_1 ≅ C_1 ≅ su(2); B_2 ≅ C_2; D_3 ≅ A_3):| Type | Lie Algebra | Rank | Dimension | Representative Group |
|---|---|---|---|---|
| A_n | su(n+1) | n | n(n+2) | SU(n+1) (e.g., SU(2) for n=1) |
| B_n | so(2n+1) | n | n(2n+1) | Spin(2n+1) (e.g., Spin(5) for n=2) |
| C_n | sp(n) | n | n(2n+1) | Sp(n) (e.g., Sp(3) for n=3) |
| D_n | so(2n) | n | n(2n-1) | Spin(2n) (e.g., Spin(6) for n=3) |
| G_2 | g_2 | 2 | 14 | G_2 |
| F_4 | f_4 | 4 | 52 | F_4 |
| E_6 | e_6 | 6 | 78 | E_6 |
| E_7 | e_7 | 7 | 133 | E_7 |
| E_8 | e_8 | 8 | 248 | E_8 |