Fact-checked by Grok 2 weeks ago

Closed-subgroup theorem

The closed-subgroup theorem, also known as Cartan's closed subgroup theorem, asserts that every closed subgroup of a Lie group is itself a Lie subgroup, inheriting a smooth manifold structure from the ambient group and thus forming an embedded submanifold.^[1] This result ensures that such subgroups possess a compatible Lie algebra and exponential map, preserving the differentiable structure essential for analyzing continuous symmetries.^[2] Proved in its general form by Élie Cartan in 1930, the theorem built upon John von Neumann's 1929 demonstration of the result specifically for the general linear group GL(n, ℝ).^[1] Cartan's proof leverages the exponential map from the Lie algebra to the group, showing that the subgroup is generated by a Lie subalgebra and is closed under the group operations, thereby confirming its smooth embedding via the inverse function theorem and local diffeomorphisms. The theorem holds profound significance in Lie theory, as it guarantees that closed subgroups—such as the orthogonal group O(n) or special orthogonal group SO(n)—are themselves Lie groups, facilitating their study in applications ranging from differential geometry to quantum mechanics and representation theory.^[2] Without this result, distinguishing abstract subgroups from those with smooth structure would complicate the classification of symmetry groups, particularly in finite-dimensional settings over the reals or complexes. It also underpins corollaries, like the fact that all closed subgroups of GL(n, ℂ) are Lie groups, underscoring the theorem's role in bridging algebraic, topological, and analytic aspects of Lie groups.^[3]

Introduction

Theorem Statement

The closed-subgroup theorem, a fundamental result in the theory of Lie groups, asserts that every closed subgroup H of a Lie group G is itself a Lie subgroup of G. More formally, let G be a Lie group over \mathbb{R} or \mathbb{C}, equipped with its manifold topology, and let H be a subgroup of G that is closed as a subset in this topology. Then H admits a unique smooth manifold structure making the inclusion H \hookrightarrow G a smooth Lie group homomorphism, with the group multiplication and inversion on H inherited from G, and H is an embedded submanifold of G. The Lie algebra \mathfrak{h} of H is then given by

\mathfrak{h} = \{ X \in \mathfrak{g} \mid \exp(tX) \in H \text{ for all sufficiently small } t \in \mathbb{R} \},

where \mathfrak{g} denotes the Lie algebra of G and \exp: \mathfrak{g} \to G is the exponential map. This theorem establishes a direct correspondence between closed subgroups and Lie subalgebras via the exponential map, thereby connecting the topological condition of closure to the differential structure of Lie groups. The result was originally proved for the special case of closed subgroups of the general linear group by John von Neumann in 1929, and extended to arbitrary Lie groups by Élie Cartan in 1930.^[4]

Historical Development

The closed-subgroup theorem traces its origins to the foundational ideas of Sophus Lie in the late 19th century, where he developed the theory of continuous transformation groups as a means to study symmetries in differential equations, establishing the framework for what would become Lie group theory.^[5] A pivotal early result came in 1929 from John von Neumann, who proved that any closed subgroup of the general linear group in a finite-dimensional real or complex vector space is itself a Lie group, thereby linking topological closure directly to the smooth structure of such subgroups.^[6] This work addressed linear cases and highlighted the importance of closure in preserving the Lie group properties.^[7] The modern formulation of the theorem for arbitrary Lie groups is attributed to Élie Cartan in the 1930s, who demonstrated that every closed subgroup of a Lie group inherits a Lie group structure as an embedded submanifold, with the smooth topology agreeing with the subspace topology; this precise statement appeared in his treatise on Lie groups around 1937. Hermann Weyl contributed to the broader context of Lie group theory during this period through his work on representations and classical groups, which influenced the understanding of subgroup structures.^[8] In the 1950s, the theorem played a central role in resolving Hilbert's fifth problem, with Andrew Gleason, Deane Montgomery, and Leo Zippin proving that topological groups locally homeomorphic to Euclidean spaces are Lie groups, and emphasizing how closed subgroups ensure the compatibility of analytic structures in such settings; their seminal 1955 monograph Topological Transformation Groups formalized these connections.^[9] While the finite-dimensional case solidified during this era, later developments by John Milnor in the 1980s extended aspects of Lie group theory, including subgroup considerations, to infinite-dimensional settings, though the core theorem remains focused on finite dimensions.^[10]

Background Concepts

Lie Groups and Subgroups

A Lie group is a mathematical structure that combines the algebraic properties of a group with the geometric properties of a smooth manifold. Specifically, it is a group G equipped with a smooth manifold structure such that the group multiplication map G \times G \to G, (g, h) \mapsto gh, and the inversion map G \to G, g \mapsto g^{-1}, are smooth maps.^[11] This framework allows for the study of continuous symmetries in differential geometry and physics, where the smoothness ensures compatibility between infinitesimal and global structures.^[12] Subgroups of a Lie group G are subsets H \subseteq G that form groups under the restriction of the multiplication and inversion operations from G. These algebraic subgroups need not be closed in the topological sense nor endowed with a smooth manifold structure inherited from G. In contrast, a Lie subgroup arises when H is itself a Lie group with the induced smooth structure, typically requiring the inclusion map H \to G to be smooth.^[13] Central to the local structure of Lie groups is the associated Lie algebra, which captures the infinitesimal behavior near the identity. The Lie algebra \mathfrak{g} of [G](/page/G) is the tangent space at the identity T_e G, viewed as a vector space over \mathbb{R} or \mathbb{C}, equipped with a Lie bracket [X, Y] defined as the commutator of the left-invariant vector fields on G corresponding to tangent vectors X, Y \in \mathfrak{g}. This bracket satisfies bilinearity, antisymmetry, and the Jacobi identity, providing a linear approximation to the nonlinear group operations.^[14] For smooth subgroups, a key distinction lies between immersed and embedded realizations. An immersed subgroup H of G has an inclusion map that is a smooth immersion—locally a diffeomorphism onto its image—but the image may be dense or fail to be a proper submanifold globally. An embedded subgroup, however, is a smooth submanifold of G, where the inclusion is both an immersion and a topological embedding, with the image being a closed submanifold of G.^[15]

Topological Closure in Lie Groups

In the context of Lie groups, the underlying topological structure ensures that the group operations—multiplication and inversion—are continuous maps with respect to the manifold topology, making every Lie group a topological group.^[16] This continuity is fundamental, as it allows the algebraic structure to interact seamlessly with the geometric properties of the manifold, enabling the study of limits and convergence within the group.^[17] A subgroup H of a Lie group G is defined as closed if it forms a closed subset of G in the topological sense, meaning the complement G \setminus H is open.^[15] In contrast, dense subgroups, whose topological closure equals the entire group G, serve as counterexamples illustrating that arbitrary subgroups do not automatically possess a Lie group structure, even if the ambient group does.^[12] Such dense subgroups highlight the necessity of closure for inheriting the smooth manifold properties and associated infinitesimal structure, like the Lie algebra.^[12] Lie groups are standardly assumed to be Hausdorff topological spaces, which guarantees the uniqueness of limits for convergent sequences and nets, preventing pathological behaviors in the topology.^[18] This Hausdorff condition is crucial for the well-definedness of the manifold structure and ensures that the group topology supports the required separation properties for analytic continuations.^[19] Finite-dimensional Lie groups over the real numbers \mathbb{R} are equipped with a natural uniform structure derived from their complete metric topology, rendering them Polish groups—separable topological groups that are completely metrizable.^[20] As Polish spaces, these Lie groups satisfy the Baire category theorem, which asserts that they cannot be expressed as countable unions of nowhere dense sets, providing a topological foundation for genericity arguments and structural theorems in the theory.^[21]

Illustrative Examples

Examples of Closed Subgroups

The special orthogonal group SO(n) provides a fundamental example of a closed subgroup within the general linear group GL(n, \mathbb{R}). It comprises all n \times n real matrices A that satisfy A^T A = I_n and \det A = 1, where I_n denotes the n \times n identity matrix; these algebraic conditions ensure SO(n) is a closed subset of GL(n, \mathbb{R}) under its standard topology induced from the space of all real matrices.^[22] The Lie algebra of SO(n), denoted \mathfrak{so}(n), consists precisely of the n \times n skew-symmetric real matrices, that is, matrices X fulfilling X^T = -X.^[23] Another illustrative case is the torus represented by the circle group U(1), which embeds as a closed subgroup of GL(2, \mathbb{R}) through the parametrization of rotation matrices:

\begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix},

where \theta ranges over [0, 2\pi). This subgroup is compact, rendering it closed in the Hausdorff topological space GL(2, \mathbb{R}).^[24] Discrete subgroups offer further examples, particularly finite groups such as cyclic groups \mathbb{Z}/k\mathbb{Z} embedded pointwise into a Lie group like GL(n, \mathbb{R}) via faithful representations (e.g., permutation matrices for suitable n \geq k). Such embeddings yield closed subgroups because finite sets are compact and thus closed in any Hausdorff space, including Lie groups.^[25] In each of these instances, the smoothness of the subgroups as Lie groups is verified through explicit manifold charts that align with the smooth structure of the ambient Lie group, consistent with the closed-subgroup theorem's assurance of an induced manifold structure.^[2]

Examples of Non-Closed Subgroups

One prominent example of a non-closed subgroup arises in the 2-torus T^2 = \mathbb{R}^2 / \mathbb{Z}^2, considered as a compact Lie group under componentwise addition modulo 1. Consider the subgroup H generated by the element (\alpha, \beta) \in T^2, where \alpha / \beta is irrational (more precisely, where 1, \alpha, \beta are linearly independent over \mathbb{Q}). This subgroup is the image of the embedding \mathbb{R} \to T^2 given by t \mapsto (t \alpha \mod 1, t \beta \mod 1), which is an immersed 1-dimensional submanifold diffeomorphic to \mathbb{R}. However, H is dense in T^2 and thus not closed, as its closure is the entire torus.^[26] A similar phenomenon occurs in the Lie group \mathrm{SL}(2, \mathbb{R}). The maximal compact subgroup \mathrm{SO}(2) consists of rotation matrices \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix} for \theta \in [0, 2\pi). The subgroup K generated by rotations through angles that are integer multiples of $2\pi \alpha, where \alpha is irrational, is \{ \begin{pmatrix} \cos (2\pi n \alpha) & -\sin (2\pi n \alpha) \\ \sin (2\pi n \alpha) & \cos (2\pi n \alpha) \end{pmatrix} \mid n \in \mathbb{Z} \}. This discrete subgroup is dense in \mathrm{SO}(2) and hence not closed in \mathrm{SL}(2, \mathbb{R}), since \mathrm{SO}(2) itself is closed but K fails to contain isolated points outside its closure.^[27] Such non-closed subgroups illustrate the necessity of the closure assumption in the closed-subgroup theorem, as they are immersed Lie subgroups but not embedded submanifolds. Consequently, they lack the smooth manifold structure required to be Lie subgroups in the standard sense, preventing the formation of a well-defined Lie algebra or tangent space at every point that aligns with the ambient group's topology.^[28] The density of these subgroups can be visualized through Weyl's equidistribution theorem: for irrational \alpha, the sequence n \alpha \mod 1 is equidistributed (and thus dense) in [0,1), implying that the orbit under repeated addition fills the torus uniformly without gaps.^[26] In contrast, closed subgroups embed as smooth submanifolds.^[27]

Properties and Extensions

Conditions for Subgroup Closure

In Lie groups, analytic subgroups—those generated by one-parameter subgroups via the exponential map from a Lie subalgebra—are always topologically closed. This follows from the fact that such subgroups inherit a smooth manifold structure compatible with the group operations and the ambient topology, ensuring no accumulation points outside the subgroup itself.^[29] Compact subgroups of Lie groups are necessarily closed. Since Lie groups are Hausdorff topological spaces, any compact subset is closed in the relative topology, and compactness prevents the subgroup from having limit points exterior to itself. This property extends the Heine-Borel theorem from Euclidean spaces to the more general locally compact setting of Lie groups.^[30] Finitely generated subgroups of nilpotent Lie groups need not be closed; dense examples exist, generated by elements whose orbits fill the group without forming a closed set. However, discrete subgroups within these groups are always closed, as the discrete topology induced on them implies no accumulation points in the ambient space. Such discrete subgroups are closed regardless of index, though those of finite covolume (lattices) play a prominent role in homogeneous space constructions.^[31] A general criterion for closure of a subgroup H in a Lie group G is that H intersects every compact subset of G in a compact set. In the \sigma-compact, locally compact framework of Lie groups, this ensures that H has no limit points outside itself, as intersections with compacta capture the topological behavior exhaustively. Equivalently, H is closed if and only if the quotient space G/H is Hausdorff, since the projection map identifies closedness with the separation of distinct cosets in the quotient topology.^[30]

Converse Results

A fundamental converse to the closed-subgroup theorem asserts that every embedded Lie subgroup of a Lie group is closed in the ambient topology. This equivalence—namely, that a Lie subgroup is embedded if and only if it is closed—holds generally for finite-dimensional Lie groups and ensures that the smooth structure induced from the embedding aligns with the intrinsic group topology.^[26] For smooth immersed Lie subgroups, the situation differs: such subgroups need not be closed, even when the ambient Lie group is connected. A classic counterexample is the irrational winding subgroup on the 2-torus \mathbb{T}^2 = \mathbb{R}^2 / \mathbb{Z}^2, generated by the one-parameter flow t \mapsto (t \mod 1, \alpha t \mod 1) where \alpha is irrational; this yields a connected immersed submanifold diffeomorphic to \mathbb{R} that is dense (hence non-closed) in \mathbb{T}^2.^[32] However, in disconnected Lie groups, additional pathologies arise, such as immersed subgroups that fail to be closed due to dense components interacting with discrete parts, exacerbating the non-closure issue beyond the connected case.^[2] A related refinement concerns analytic subgroups, defined as subgroups that are real-analytic submanifolds of the ambient Lie group. The Montgomery–Zippin theorem, building on Hilbert's fifth problem, implies that every analytic subgroup of a finite-dimensional Lie group is closed, as such subgroups inherit a unique Lie group structure compatible with the ambient topology and must coincide with their topological closure. This result underscores the stronger regularity provided by analyticity over mere smoothness.^[33] These converses fail in the infinite-dimensional setting without supplementary topological assumptions, such as completeness of the model space (e.g., Banach or Fréchet Lie groups). For instance, in separable Hilbert Lie groups, there exist closed subgroups that are not Lie subgroups, and conversely, immersed subgroups that remain non-closed despite satisfying finite-dimensional analogs of immersion conditions. Additional structure, like the interaction of the exponential map with the subgroup topology, is required to recover closure properties.^[34]

Applications

In Representation Theory

In the context of unitary representations of compact Lie groups, the closed-subgroup theorem guarantees that any closed subgroup inherits the smooth structure of a Lie group, enabling the Peter-Weyl theorem to decompose unitary representations into orthogonal direct sums of finite-dimensional irreducible components. This decomposition is essential for analyzing how representations of the ambient group restrict to closed subgroups, preserving irreducibility or allowing controlled branching rules. For instance, the representation ring of a closed subgroup H of a compact Lie group G embeds into that of G via induction, facilitating the classification of representations through subgroup data. The theorem also plays a pivotal role in the construction of induced representations. For a closed subgroup H of a Lie group G and a smooth representation \pi of H, the induced representation \operatorname{Ind}_G^H \pi is smooth, ensuring that the resulting action on functions over the quotient space G/H is continuous and differentiable. This smoothness property is critical for extending representation-theoretic techniques from subgroups to the full group, particularly in non-compact settings where topological closure prevents pathological behaviors.^[26] Mackey theory provides a deeper connection, where the closed-subgroup theorem underpins the generalization of Frobenius reciprocity to Lie groups. In this framework, closedness of H ensures that the intertwining number between \operatorname{Ind}_G^H \pi and a representation of G equals the dimension of invariants under \pi, allowing reciprocity between induction from H and restriction to H. This result extends classical finite-group reciprocity, relying on the Lie group structure afforded by closure.^[35] A concrete illustration arises in the special unitary group \operatorname{SU}(2), where the maximal torus U(1) of diagonal matrices forms a closed subgroup. Inducing characters of U(1) to \operatorname{SU}(2) produces all finite-dimensional irreducible representations of \operatorname{SU}(2), each of odd dimension $2\ell + 1 for \ell \in \mathbb{N}_0, highlighting how closure enables explicit construction via highest weights.^[36]^[37]

In Homogeneous Spaces

In the context of Lie groups, the closed-subgroup theorem ensures that if H is a closed subgroup of a Lie group G, then the quotient space G/H inherits a natural smooth manifold structure, with dimension equal to \dim G - \dim H. This structure arises because the canonical projection \pi: G \to G/H becomes a smooth submersion, allowing G/H to be endowed with a differential structure compatible with the group action. Moreover, the existence of this smooth structure facilitates the integration of functions on G/H using the Haar measure on G, as the measure class on the quotient is well-defined precisely when H is closed, enabling disintegration theorems and invariant measures on homogeneous spaces.^[38] The closedness of H also guarantees that the projection \pi: G \to G/H is a principal H-bundle, ensuring local triviality in associated fiber bundle constructions. Specifically, the left action of H on G by multiplication is free and proper when H is closed, making \pi: G \to G/H a principal H-bundle with smooth total space and base. This property underpins the smooth structure of associated vector bundles and connections over G/H, which are crucial for geometric analysis on these spaces.^[39]^[40] Symmetric spaces provide a key application where closed subgroups play a central role. A Riemannian symmetric space can be realized as G/H, where G is a Lie group acting by isometries, and H is the closed centralizer of an involutive automorphism \sigma of G (i.e., \sigma^2 = \mathrm{id} and \sigma \neq \mathrm{id}). The fixed-point subgroup H = \{g \in G \mid \sigma(g) = g\} is closed (as the fixed points of a continuous automorphism), and hence a Lie subgroup by the closed-subgroup theorem, ensuring G/H is a smooth manifold equipped with a G-invariant Riemannian metric for which the geodesic symmetries are isometries. A canonical example is the space of positive definite matrices \mathrm{SL}(n,\mathbb{R})/\mathrm{SO}(n), which models hyperbolic geometry and arises from the involution transposing matrices.^[41] In the study of Lie group actions on manifolds, the closed-subgroup theorem implies that stabilizers of points under smooth actions are closed Lie subgroups, leading to smooth orbit structures. For a smooth action of G on a manifold M, the stabilizer \mathrm{Stab}_G(x) of a point x \in M is closed, hence a Lie subgroup, and the orbit G \cdot x is diffeomorphic to G / \mathrm{Stab}_G(x), inheriting a smooth immersed submanifold structure. When the action is proper, closed stabilizers further ensure that orbits are embedded submanifolds, classifying them as smooth components in the orbit decomposition of M. This classification is essential for understanding foliations and reduction in symplectic geometry on such spaces.^[39]

Proof Outline

Key Technical Lemma

A central component in establishing the closed-subgroup theorem is a lemma describing the local behavior of a closed subgroup H near the identity element e of the ambient Lie group G. Let \mathfrak{g} denote the Lie algebra of G and \mathfrak{h} the Lie algebra of H, defined as \mathfrak{h} = \{ X \in \mathfrak{g} \mid \exp(tX) \in H \ \forall t \in \mathbb{R} \}. The lemma asserts that there exists a neighborhood U of e in G such that

H \cap U = \exp(\mathfrak{h}) \cap U,

where \exp: \mathfrak{g} \to G is the exponential map.^[42] The exponential map \exp: \mathfrak{g} \to G is smooth, and its differential at the origin d\exp_0: \mathfrak{g} \to T_e G \cong \mathfrak{g} is the identity isomorphism. By the inverse function theorem, \exp is a local diffeomorphism: there are neighborhoods W of $0 in \mathfrak{g} and V of e in G such that \exp restricts to a diffeomorphism W \to V. Moreover, \exp is surjective onto a neighborhood of e within the connected component of the identity in G. The proof of the lemma relies on a tubular neighborhood construction or slice theorem via a direct sum decomposition \mathfrak{g} = \mathfrak{h} \oplus \mathfrak{m}, where \mathfrak{m} is a complementary subspace. Define \Phi: \mathfrak{g} \to G by \Phi(X + Y) = \exp(X) \exp(Y) for X \in \mathfrak{h}, Y \in \mathfrak{m}; this map is a diffeomorphism onto its image near e, with d\Phi_0 the identity. To show the equality, assume for contradiction a sequence g_n \in H \cap V with g_n \to e but g_n \notin \exp(\mathfrak{h}). Writing g_n = \exp(X_n) \exp(Y_n) with Y_n \in \mathfrak{m} \setminus \{0\} and X_n + Y_n \to 0, then \exp(-X_n) g_n = \exp(Y_n) \in H. For large n, Y_n is small, so Y_n = \log(\exp(Y_n)) \in \mathfrak{h}, contradicting \mathfrak{h} \cap \mathfrak{m} = \{0\}. Thus, near e, elements of H arise solely from \exp(\mathfrak{h}). This local equality, combined with the diffeomorphism property of \exp restricted to \mathfrak{h}, yields the desired result.^[42]^[43]

Core Proof Argument

The core proof of the closed-subgroup theorem proceeds by establishing that a closed subgroup H of a Lie group G with Lie algebra \mathfrak{g} inherits a smooth manifold structure and Lie group operations from G, building on a local technical lemma that provides a neighborhood of the identity in H diffeomorphic to an open subset of a vector space.^[42] To show the global structure, the local diffeomorphism near the identity—given by the exponential map restricted to a subspace—is extended across H using left translations by elements of H. Specifically, for any h \in H, the left translation \lambda_h: G \to G given by \lambda_h(g) = h g maps the local chart neighborhood in H to a neighborhood of h, yielding an atlas of charts that covers H and renders the inclusion H \hookrightarrow G a smooth embedding of submanifolds. This construction ensures H is a smooth submanifold of G of dimension equal to that of the local model.^[42]^[43] The Lie algebra \mathfrak{h} of H is identified as the subspace

\mathfrak{h} = \{ X \in \mathfrak{g} \mid \exp(tX) \in H \ \forall t \in \mathbb{R} \text{ sufficiently small} \},

where \exp: \mathfrak{g} \to G is the exponential map. This set \mathfrak{h} is a vector subspace of \mathfrak{g} and is closed under the Lie bracket [\cdot, \cdot], as the one-parameter subgroups generated by [X, Y] for X, Y \in \mathfrak{h} remain in H by the Baker-Campbell-Hausdorff formula up to higher-order terms.^[42]^[43] The group operations on H are smooth as restrictions of those on G: the multiplication map m: H \times H \to H, m(h_1, h_2) = h_1 h_2, is the restriction of G's smooth multiplication, and similarly for the inverse map i: H \to H, i(h) = h^{-1}, which inherits smoothness from G's inversion. These restrictions are smooth because H is a smooth submanifold embedded in G.^[42]^[44] For the connected case, the proof first establishes the structure on the identity component H_0 using the above local-global extension, then handles the full H by noting that its connected components are the cosets K H_0 for finitely many coset representatives K (since H is second countable), each of which inherits smoothness as a translate of the submanifold H_0, forming a smooth disjoint union.^[42] Finally, the adjoint action of G preserves \mathfrak{h}, as conjugation by g \in G maps one-parameter subgroups in H to those in H: specifically,

\exp(t \operatorname{Ad}_g X) = g \exp(t X) g^{-1} \in H

for X \in \mathfrak{h}, implying \operatorname{Ad}_g X \in \mathfrak{h}. This confirms the compatibility of \mathfrak{h} with G's structure.^[42]^[43]

References

[1]
[PDF] Notes on compact Lie groups - ETH Zürich
Oct 17, 2022 · The following theorem was first proved in 1929 by John von Neumann [5] for the special case G = GL(n,R) and then in 1930 by Élie Cartan [1] in ...Missing: original | Show results with:original
[2]
[PDF] THE CLOSED SUBGROUP THEOREM Contents 1. Lie Groups 1 2 ...
It states that any closed subgroup of a Lie group is itself a Lie group, and therefore inherits a smooth manifold structure.
[3]
[PDF] 2 Lie groups and algebraic groups. - UCSD Math
Theorem 3 A closed subgroup of GL(n;C) is a Lie group. This theorem is a special case of the fact that a closed subgroup of a. Lie group is a Lie group. We ...<|control11|><|separator|>
[4]
Über die analytischen Eigenschaften von Gruppen linearer ...
Über die analytischen Eigenschaften von Gruppen linearer Transformationen und ihrer Darstellungen ... Article PDF. Download to read the full article text.Missing: linearen | Show results with:linearen
[5]
None
### Summary of Historical Development of Lie Theory
[6]
[PDF] JOHN VON NEUMANN - National Academy of Sciences
In 1929 he published the important result (# 24) that a group of ... that a closed subgroup of a Lie group is again a Lie group, and he proved it ...
[7]
[PDF] Introduction to Lie groups
The following theorem was proven by John von Neumann in 1929. 4.2.4 ... Lie groups: any closed subgroup of a Lie group is a closed Lie subgroup. To.
[8]
[PDF] Compact Lie Groups - University of Oregon
May 5, 2022 · These notes are about Lie groups which are compact, and about the irreducible represen- tations of these groups. In our sketch of the ...<|control11|><|separator|>
[9]
Gleason — Palais - Celebratio Mathematica
This led in 1952 to a satisfying denouement to the story of the Fifth Problem, with Gleason and Montgomery–Zippin carrying out the above two-pronged attack.
[10]
Fundamental problems in the theory of infinite-dimensional Lie groups
Feb 5, 2006 · John Milnor formulated various fundamental questions concerning infinite-dimensional Lie groups. In this note, we describe some of the answers.Missing: subgroups 1980s
[11]
Lie group in nLab
### Definition of a Lie Group
[12]
[PDF] Introduction to Lie Groups and Lie Algebras Alexander Kirillov, Jr.
Finally, there is an analog of the standard homomorphism theorem for Lie groups. ... Without using the theorem about closed subgroup, show that H is a Lie group.
[13]
None
### Summary of Lie Subgroup Definitions from Lec 10.pdf
[14]
[PDF] Lie groups and Lie algebras (Winter 2024)
By definition, matrix Lie groups are embedded submanifolds of GL(n,R), hence a closed subgroup of a matrix Lie group is a closed subgroup of GL(n,R). Hence it ...
[15]
[PDF] lecture 18: lie subgroups
Definition 1.2. A subgroup H of a Lie group G is called a Lie subgroup if it is an immersed submanifold, and the group multiplication µH = µG|H×H is smooth.
[16]
[PDF] Topological Groups in Optimization - Michael Orlitzky
In particular µ and ι are smooth operations, and are thus continuous. So every Lie group is a topological group. Example. Aut(K), the automorphism group of.
[17]
[PDF] An Introduction to Topological Groups - Carleton University
Jan 7, 2021 · Theorem 2.2.5. If G is a topological group, then {e} is a closed normal subgroup of G and its topology is the indiscrete topology.<|separator|>
[18]
[PDF] Lie Groups: Fall, 2022 Lecture I - Columbia Math Department
Aug 24, 2022 · Usually, manifolds are assumed to be Hausdorff and second countable. ... Most Lie groups, complex Lie groups, or linear algebraic groups ...
[19]
Representations of Lie groups - arXiv
This course will be about representations of Lie groups, with a focus on non-compact groups. ... 4Topological groups will always be assumed Hausdorff and second ...
[20]
Uniqueness of Polish group topology - ScienceDirect.com
It follows that compact, connected, simple Lie groups, as well as finitely generated profinite groups, have a unique Polish group topology.
[21]
[PDF] Polish groups and Baire category methods - Numdam
This article explores Polish groups, Baire category methods, and metric model theory, using Baire category as a substitute for measure-theoretic concepts.
[22]
[PDF] 2.5 Lie Groups
so SO(n) is an ... Let G C GL(n, R) be a subgroup. Prove that G is a Lie group if and only if it is a closed subset of GL(n, R) in the relative topology.
[23]
[PDF] Matrix Lie groups and their Lie algebras - Alen Alexanderian
Jul 12, 2013 · An equivalent way of definiting matrix Lie groups is to define them as closed subgroups of GL(n). 4.1 Examples. Let us look at some examples of ...
[24]
[PDF] Rotations and reflections in the plane - Purdue Math
The set of orthogonal matrices O(2) forms a subgroup of GL2(R). Given a unit vector v 2 R2 and A 2 O(2), Av is also a unit vector. So we can interpret ...
[25]
Can discrete groups be Lie groups? Are all finite ... - MathOverflow
Jan 20, 2022 · It appears to me that discrete groups can be Lie groups, and all finite groups are Lie groups. This contradicts what I thought Lie groups were.Finite maximal closed subgroups of Lie groups - MathOverflowDiscrete central subgroup of a connected Lie group is finitely ...More results from mathoverflow.net
[26]
[PDF] Lie Groups. Representation Theory and Symmetric Spaces
In this Chapter we discuss elementary properties of Lie groups, Lie algebras and their relationship. We will assume a good knowledge of manifolds, vector.<|control11|><|separator|>
[27]
[PDF] An Introduction to Lie Groups and Symplectic Geometry
Jul 23, 2018 · These are the lecture notes for a short course entitled “Introduction to Lie groups and symplectic geometry” that I gave at the 1991 ...<|control11|><|separator|>
[28]
None
Summary of each segment:
[29]
[PDF] Lie Groups Beyond an Introduction
... analytic subgroups are closed and simply connected, and we are interested in the analytic subgroup Z corresponding to z = nn−1. Corollary 1.126 notes that ...
[30]
[PDF] Part I Basic Properties of Topological Groups
May 19, 2020 · The quotient G/H is Hausdorff if and only if H is closed in G, and it is discrete if and only if H is open in G. Proof. Suppose that U ⊆ G is an ...
[31]
[PDF] equidistribution of dense subgroups on nilpotent lie groups
Our equidistribution problem of a dense subgroup Γ in a nilpotent Lie group G can be ... connected nilpotent Lie group and Γ a finitely generated torsion-free ...
[32]
[PDF] Chapter 5 Lie Groups, Lie Algebras and the Exponential Map
Observe that an immersed Lie subgroup, H, is an im- mersed submanifold, since ϕ is an injective immersion. However, ϕ(H) may not have the subspace topology in-.
[33]
THREE NOTES ON REAL ANALYTIC GROUPS - Project Euclid
Let G be a real Lie group and A an analytic subgroup of G. Our objective is to describe the closure B = A of A in G. The basic results are due to Malcev [6].
[34]
[PDF] Infinite-Dimensional Lie Theory for Gauge Groups
Oct 20, 2006 · Since an infinite-dimensional Lie group may posses closed subgroups ... dimensional Lie theory has been done by John Willard Milnor in [Mi84],.
[35]
[PDF] Chapter 4 Analysis on Compact Groups and Representations
In this section we prove the second part of the Peter–Weyl theorem which has to do with unitary representations. ... closed subgroup of G) we use the notation R(G) ...
[36]
[PDF] Introduction to Lie Groups and Lie Algebras Alexander Kirillov, Jr.
This book is an introduction to the theory of Lie groups and Lie algebras, with emphasis on the theory of semisimple Lie algebras.
[37]
Induced Representations of Locally Compact Groups II. The ... - jstor
groups Theorem 4.1 is equivalent to the Frobenius reciprocity theorem and in this sense is a generalization of that theorem to non compact groups. The.Missing: Mack | Show results with:Mack
[38]
[PDF] Math 210B. Mackey theory and applications
Serre's book Linear representations of finite groups via Mackey's criterion and Frobenius reciprocity that the irreducible representations of G are given ...Missing: Mack | Show results with:Mack
[39]
[PDF] Representation Theory And Quantum Mechanics
Therefore, if we have a representation π of SU(2) π : SU(2) → GL(n,C) we can restrict π to this U(1) subgroup and turn π into a representation of U(1).
[40]
[PDF] Quantum Theory, Groups and Representations: An Introduction ...
For compact Lie groups, emphasis is on the groups U(1),SO(3),SU(2) and their finite dimensional repre- sentations. Central to the basic structure of quantum ...
[41]
[PDF] 4. Homogeneous spaces, Lie group actions - MIT OpenCourseWare
Theorem 4.1. (i) Let G be a Lie group of dimension n and H ⊂ G a closed Lie subgroup of dimension k. Then the homogeneous space. G/H has a natural structure ...
[42]
[PDF] Group actions on manifolds | Lecture Notes
It is important to note that the Theorem becomes false if we consider principal bundles in the category of topological spaces. In the definition of a ...
[43]
[PDF] Notes on principal bundles and classifying spaces
Call a subgroup H of G admissible if the quotient map G −→ G/H is a principal H-bundle. For example, any subgroup of a discrete group is admissible, and any ...
[44]
[PDF] SYMMETRIC SPACES - Penn Math
Jan 30, 2025 · A Riemannian symmetric space is a manifold where the geodesic symmetry around every point is an isometry.
[45]
[PDF] Hilbert's Fifth Problem and Related Topics - Terry Tao
The theorems of Cartan and von Neumann. We now turn to the proof of Cartan's theorem. As indicated in the introduction, the fundamental concept here will be ...<|separator|>
[46]
None
### Summary of Core Proof from ClosedLie.pdf
[47]
None
### Summary of Core Proof Steps for Closed Subgroup Theorem (Rutgers Notes)