Fact-checked by Grok 2 weeks ago

Lie group

A Lie group is a mathematical structure that is simultaneously a group and a smooth manifold, with the group operations of multiplication and inversion required to be smooth maps between manifolds.^[1] This compatibility allows Lie groups to model continuous symmetries in a geometrically natural way, where elements represent transformations and the manifold structure encodes their differentiable nature.^[2] Finite-dimensional Lie groups, the primary focus of classical theory, include familiar examples such as the general linear group GL(n, ℝ) of invertible n×n real matrices and the orthogonal group O(n) preserving the Euclidean inner product.^[3] The theory of Lie groups originated in the late 19th century with the work of Norwegian mathematician Sophus Lie (1842–1899), who developed it to classify continuous groups of transformations as a tool for solving ordinary differential equations through symmetry methods. Lie's ideas were later formalized and expanded in the 20th century by mathematicians like Hermann Weyl and Élie Cartan, who connected Lie groups to their associated Lie algebras—vector spaces at the identity element capturing infinitesimal group structure via the Lie bracket.^[4] Semisimple Lie groups, classified up to isomorphism by their root systems (as in the Cartan-Killing classification), form a cornerstone of the subject, with compact forms like SU(n) and SO(n) playing central roles in representation theory.^[1] Lie groups underpin modern applications across mathematics and physics, from describing symmetries in general relativity and quantum field theory to enabling algorithms in robotics and computer vision.^[5] In particle physics, they model gauge symmetries, such as the SU(3) color group in quantum chromodynamics.^[6] Their interplay with differential geometry facilitates the study of homogeneous spaces and invariant metrics, while in control theory, Lie group methods optimize trajectories on non-Euclidean configuration spaces.^[7]

Historical Development

Origins and Early Ideas

The origins of Lie groups emerged from 19th-century advancements in geometry and algebra, where mathematicians sought to generalize discrete symmetry groups to continuous ones, enabling the analysis of symmetries in differential equations and geometric structures. Early influences included Felix Klein's 1872 Erlangen Program, which classified geometries based on their underlying groups of transformations, emphasizing symmetry as a unifying principle across geometric theories. Henri Poincaré further contributed in the 1880s through his studies of symmetry groups in automorphic functions and differential equations, highlighting continuous transformations in the context of complex analysis and celestial mechanics.^[8] Sophus Lie, building on these geometric insights, pioneered the development of infinitesimal transformations during the 1870s and 1880s, conceptualizing continuous groups of transformations as tools for integrating partial differential equations. His approach marked a pivotal shift from discrete groups, like those studied by Galois, to continuous symmetries, where transformations could vary smoothly and be parameterized by real or complex numbers, thus capturing infinitesimal changes in geometric configurations.^[9] A key milestone was Lie's 1873 paper "Über die Kontakttransformationen," which explored contact transformations preserving the tangency of curves and surfaces, demonstrating how such continuous symmetries could classify solutions to differential equations.^[10] This work, motivated by problems in projective geometry and influenced by Klein's ideas on invariant theory, laid the groundwork for viewing transformation groups as acting locally and infinitesimally on manifolds.^[11] The pre-20th-century timeline progressed with Lie's systematic classification efforts in the 1880s, including his 1888-1893 three-volume treatise Theorie der Transformationsgruppen, co-authored with Friedrich Engel, which formalized continuous groups through their infinitesimal generators.^[4] Élie Cartan extended these foundations in the 1890s, notably in his 1894 doctoral thesis Sur la structure des groupes de transformations finis et continus, where he analyzed the structure of continuous transformation groups using moving frames and involutory systems, bridging Lie's geometric methods with algebraic insights. These developments set the stage for the transition to Lie algebras in the early 20th century.

Key Contributions and Evolution

Élie Cartan made foundational contributions to Lie group theory through his doctoral thesis in 1894, titled "Sur la structure des groupes de transformations finis et continus," where he provided a complete classification of simple Lie algebras over the complex numbers, building on Wilhelm Killing's earlier work and integrating the theory with differential geometry by examining the structure equations of Lie algebras.^[12] In his subsequent publications from 1899 to 1905, Cartan further advanced this integration by developing the method of moving frames, a technique that adapts local coordinate systems along manifolds under the action of Lie groups, enabling the study of geometric invariants and equivalence problems in differential geometry. Hermann Weyl significantly expanded the theory in 1925 through a series of papers on the representation theory of compact Lie groups, introducing methods to decompose representations into irreducible components using characters and highest weights, which provided a systematic framework for understanding symmetries in physical systems.^[13] Weyl applied these ideas to quantum mechanics in his 1928 book Gruppentheorie und Quantenmechanik^[14] and subsequent works, emphasizing unitary representations of Lie groups to describe the symmetry groups underlying atomic spectra and wave functions, thereby bridging abstract group theory with the emerging principles of quantum theory. Following World War II, Claude Chevalley introduced an algebraic perspective on Lie groups in his 1946 book "Theory of Lie Groups," shifting focus from analytic and topological aspects to purely algebraic structures, the use of algebraic methods to prove key theorems on the topology and representations of compact Lie groups.^[15] This algebraic approach complemented the earlier Cartan-Killing classification of semisimple Lie algebras, which identifies all such algebras up to isomorphism via their root systems and Dynkin diagrams, a framework that gained renewed prominence in post-war algebraic geometry and number theory for classifying semisimple groups over arbitrary fields.^[16] Lie group theory evolved into pivotal applications in particle physics during the 1960s, most notably with the adoption of the SU(3) symmetry group in the quark model proposed independently by Murray Gell-Mann in his 1964 paper "A Schematic Model of Baryons and Mesons" and George Zweig in a 1964 CERN report, where SU(3) flavor symmetry organized the spectrum of hadrons into multiplets, predicting the existence of quarks as fundamental constituents and explaining strong interaction patterns. This application demonstrated the power of Lie groups in modeling gauge symmetries, influencing the development of the Standard Model with groups like SU(3)_c for quantum chromodynamics. In the post-1980s era, computational advancements addressed longstanding challenges in explicit calculations for Lie groups, with software systems enabling practical implementations of representation theory and structure computations that were previously infeasible by hand; for instance, the LiE package, first released in 1992, facilitates computations of weights, characters, and branching rules for representations of semisimple Lie groups over finite fields. Similarly, the GAP system, through its packages like the Lie algebra package introduced in the 1990s, supports algorithmic classification and computation of Lie group structures, filling gaps in manual verification and extending applications to computational algebra and physics simulations.

Introductory Overview

Core Intuition

Lie groups represent a profound fusion of algebra and geometry, capturing continuous symmetries in a rigorous yet intuitive framework. At their core, a Lie group is a set equipped with a group structure—meaning it has an operation for combining elements (multiplication) and an inverse operation, satisfying associativity and identity properties—while also being a smooth manifold, a space that locally resembles ordinary Euclidean space and allows for seamless differentiation. The key innovation is that these group operations are themselves smooth, meaning they vary continuously and differentiably with respect to the manifold's coordinates. This setup generalizes discrete symmetries, like the finite rotations of a square (parameterized by 0°, 90°, 180°, or 270°), to infinite, continuous families, such as all possible rotations in two-dimensional space parameterized by a single angle θ ranging from 0 to 360° (or 0 to 2π radians).^[17]^[18] This continuous parameterization distinguishes Lie groups from their finite counterparts, enabling the study of symmetries that evolve gradually rather than in jumps. For instance, just as finite groups describe rigid, discrete transformations, Lie groups model fluid motions like rotations or translations, where nearby elements in the group correspond to infinitesimally small changes. The smoothness requirement is crucial because it permits the application of calculus: one can differentiate group operations to uncover "infinitesimal generators," which are tangent vectors at the identity element that generate the group's structure through flows, much like velocity vectors describe paths in physics. Without smoothness, such local approximations would fail, and the deep connection to differential equations—central to Lie's original vision—would be lost.^[19]^[20] To appreciate this intuition, basic familiarity with groups (sets closed under an associative operation with inverses and identity) and manifolds (patchwork of charts making the space differentiable) is helpful, though the essence lies in viewing Lie groups as "symmetry machines" where geometry dictates how symmetries compose. This perspective originated in the late 19th century with Sophus Lie's work on continuous transformation groups, aiming to classify symmetries via differential equations.^[21]^[22]

Importance and Applications

Lie groups play a fundamental role in understanding symmetries in physics, where continuous symmetries of physical systems correspond to Lie group actions, linking them to conservation laws via Noether's theorem. In 1918, Emmy Noether established that every differentiable symmetry of the action of a physical system, represented by a Lie group, yields a corresponding conservation law, such as energy conservation from time-translation invariance or momentum conservation from spatial translation invariance.^[23] This connection has profoundly influenced modern physics, providing a rigorous framework for analyzing invariant properties in Lagrangian mechanics.^[24] In differential geometry, Lie groups underpin the study of curvature and connections on manifolds, serving as structure groups for principal bundles that model gauge theories and geometric invariants.^[25] For instance, the frame bundle of a Riemannian manifold uses the orthogonal group O(n) as its structure group, enabling the definition of Levi-Civita connections and curvature tensors. In quantum mechanics, unitary Lie groups U(n) act on Hilbert spaces to preserve inner products, representing time evolution and symmetry transformations that maintain the probabilistic interpretation of wave functions.^[26] Lie groups also find extensive applications in engineering and applied sciences. In robotics, configuration spaces for rigid body motions are modeled by Lie groups like SE(3), facilitating kinematic modeling and path planning for manipulators and mobile robots.^[27] In computer vision, the special orthogonal group SO(3) is crucial for pose estimation, where rotations are optimized on the manifold to align 3D models with 2D images in tasks like object tracking.^[28] Modern advancements include machine learning, where Lie group neural networks, developed since the 2010s, incorporate group invariances for tasks like skeleton-based action recognition, improving generalization by respecting rotational symmetries.^[29] In chemistry, Lie groups describe continuous molecular symmetries, such as rotational groups in vibrational spectroscopy, aiding the analysis of non-rigid molecules and electronic states.^[30] Additionally, in control theory, Lie groups enhance simultaneous localization and mapping (SLAM) algorithms by parameterizing poses in SE(3), enabling robust state estimation in autonomous systems despite nonlinear constraints.^[31]

Formal Definitions

Topological Group Structure

A Lie group begins with the structure of a topological group, where the group is equipped with a topology making the multiplication map m: G \times G \to G, defined by (g, h) \mapsto gh, and the inversion map i: G \to G, defined by g \mapsto g^{-1}, both continuous.^[32] This ensures that the group operations respect the topology, allowing concepts like convergence and compactness to interact meaningfully with the algebraic structure.^[33] For the Lie group to qualify as such in the topological sense, the underlying space G must also be a topological manifold, meaning it is a Hausdorff, second-countable, locally Euclidean topological space.^[34] The Hausdorff condition separates distinct points, preventing pathological overlaps in limits, while second-countability provides a countable basis for the topology, ensuring the space is metrizable and paracompact—properties crucial for embedding theorems and partition of unity constructions in manifold theory.^[35] These assumptions guarantee that G behaves well as both a group and a space, facilitating the study of continuous homomorphisms and quotients.^[32] A classic non-example illustrates the necessity of the manifold structure: the rational numbers \mathbb{Q} under addition, endowed with the subspace topology from \mathbb{R}, form a topological group since addition and inversion are continuous restrictions from \mathbb{R}.^[36] However, \mathbb{Q} fails to be a manifold because it is totally disconnected and lacks the local Euclidean neighborhoods required, with every point having a basis of clopen sets that are neither open balls nor homeomorphic to \mathbb{R}^n intervals.^[36] This topological framework assumes familiarity with basic concepts from topology, such as manifolds and continuity of maps between spaces, but highlights that mere topological continuity of group operations is insufficient for the full Lie group theory, which requires additional smoothness to enable differential calculus.^[32]

Smooth Manifold Requirement

A Lie group G is required to possess a smooth manifold structure in addition to its topological group structure, ensuring that the group multiplication and inversion operations are infinitely differentiable (C^\infty-smooth) maps. Specifically, the multiplication map m: G \times G \to G, defined by (g, h) \mapsto gh for all g, h \in G, must be smooth at every point, and the inversion map i: G \to G, defined by g \mapsto g^{-1}, must likewise be smooth. This compatibility between the group operations and the manifold structure means that in local charts, the group laws appear as smooth functions from \mathbb{R}^n \times \mathbb{R}^n to \mathbb{R}^n, where n = \dim G.^[37] The smooth manifold requirement implies that left and right translations are diffeomorphisms. For fixed g \in G, the left translation L_g: G \to G, given by L_g(h) = gh, and the right translation R_g: G \to G, given by R_g(h) = hg, are both smooth bijections with smooth inverses, preserving the differentiable structure across the group. These translations ensure that the smooth structure is invariant under the group action on itself, allowing calculus to be performed uniformly on G.^[38]^[39] The insistence on infinite differentiability (C^\infty) rather than merely finite smoothness is crucial, as it enables the application of Taylor series expansions to approximate group elements near the identity, which is foundational for analyzing infinitesimal transformations and deriving the associated Lie algebra. This level of regularity supports higher-order derivatives essential for the local study of the group near the identity element.^[1]

Basic Examples

Classical Matrix Groups

The classical matrix Lie groups provide foundational examples of finite-dimensional Lie groups, realized as closed subgroups of the general linear group \mathrm{GL}(n, \mathbb{R}), the group of invertible n \times n real matrices under matrix multiplication. The general linear group \mathrm{GL}(n, \mathbb{R}) itself consists of all nonsingular n \times n real matrices and forms an open subset of the space M_n(\mathbb{R}) of all n \times n real matrices, endowed with the subspace topology from \mathbb{R}^{n^2}.^[40]^[41] This open set structure ensures that \mathrm{GL}(n, \mathbb{R}) is a smooth manifold of dimension n^2, with group operations of multiplication and inversion being smooth maps, making it a prototypical Lie group.^[40]^[42] Key subclasses arise by imposing additional constraints that preserve the group structure. The orthogonal group \mathrm{O}(n) comprises all n \times n real matrices g satisfying g^T g = I, where I is the identity matrix and g^T denotes the transpose; these matrices preserve the Euclidean inner product and form a closed subgroup of \mathrm{GL}(n, \mathbb{R}) under multiplication and inversion.^[43]^[42] The special orthogonal group \mathrm{SO}(n) is the connected component of \mathrm{O}(n) containing the identity, defined by the additional condition \det(g) = 1, and consists of proper rotations in \mathbb{R}^n.^[43]^[44] Both \mathrm{O}(n) and \mathrm{SO}(n) are compact Lie groups of dimension n(n-1)/2, as the orthogonality condition imposes n(n+1)/2 independent equations on the n^2 matrix entries, leaving the specified degrees of freedom; for instance, \dim(\mathrm{SO}(3)) = 3, parameterizing rotations in three-dimensional space.^[44]^[45] Over the complex numbers, analogous groups are defined within \mathrm{GL}(n, \mathbb{C}). The unitary group \mathrm{U}(n) consists of all n \times n complex matrices g satisfying g^* g = I, where g^* is the conjugate transpose (Hermitian adjoint), preserving the Hermitian inner product and forming a compact Lie subgroup closed under multiplication and inversion.^[46]^[47] The special unitary group \mathrm{SU}(n) is the kernel of the determinant map on \mathrm{U}(n), defined by \det(g) = 1, and has dimension n^2 - 1.^[46]^[47] These matrix groups are prototypical because they are embedded as closed subsets of \mathrm{GL}(n, \mathbb{R}) or \mathrm{GL}(n, \mathbb{C}), ensuring the group operations remain smooth and compatible with the manifold structure.^[42]^[45] In contrast, discrete subgroups of these matrix groups, such as finite rotation groups (e.g., the cyclic or dihedral groups embedded in \mathrm{SO}(n)), do not qualify as Lie groups in the classical sense, as they lack the positive-dimensional smooth manifold structure required for the infinitesimal analysis central to Lie theory.^[48]^[49]

One- and Two-Dimensional Cases

Up to isomorphism, the connected one-dimensional Lie groups are the additive group \mathbb{R} (non-compact and simply connected) and the circle group S^1 (compact).^[50] The circle group S^1 consists of complex numbers z \in \mathbb{C} with |z| = 1, forming an abelian group under multiplication.^[51] This group can be parameterized by an angle \theta \in [0, 2\pi), where each element is represented as e^{i\theta}, and the group operation corresponds to multiplication: e^{i\theta} \cdot e^{i\phi} = e^{i(\theta + \phi \mod 2\pi)}.^[52] The abelian nature arises because multiplication is commutative in this representation, reflecting the underlying addition of angles modulo $2\pi.^[51] The circle group S^1 is isomorphic to the quotient group \mathbb{R}/\mathbb{Z}, where the identification scales the period to 1, providing an additive structure equivalent to the angular parameterization.^[53] It is also isomorphic to the special orthogonal group SO(2), consisting of $2 \times 2 rotation matrices, and to the unitary group U(1) of $1 \times 1 unitary matrices, all sharing the same Lie group structure as the manifold \mathbb{R}/2\pi\mathbb{Z}.^[54] Geometrically, S^1 is visualized as a one-dimensional torus (circle), with geodesics being arcs of constant speed along the circle, illustrating the compact, connected topology.^[54] In two dimensions, a fundamental non-abelian example is the affine group, realized as the group of $2 \times 2 upper triangular real matrices of the form

\begin{pmatrix} a & b \\ 0 & 1 \end{pmatrix},

where a > 0, b \in \mathbb{R}, under matrix multiplication.^[54] The group operation is given by

\begin{pmatrix} a & b \\ 0 & 1 \end{pmatrix} \begin{pmatrix} a' & b' \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} a a' & a b' + b \\ 0 & 1 \end{pmatrix},

revealing non-commutativity through the term a b' in the off-diagonal component.^[55] This group is solvable, as its derived series terminates (the commutator subgroup is the center, isomorphic to \mathbb{R}), distinguishing it from abelian two-dimensional groups like the torus S^1 \times S^1.^[54] Geometrically, the affine group acts on \mathbb{R} by dilations and translations, providing intuition for its non-abelian structure in the context of one-dimensional affine transformations.

Lie Algebras

Definition and Construction

The Lie algebra \mathfrak{g} of a Lie group G is defined as the tangent space T_e G at the identity element e \in G, equipped with a bilinear operation known as the Lie bracket [\cdot, \cdot]: \mathfrak{g} \times \mathfrak{g} \to \mathfrak{g} that satisfies antisymmetry and the Jacobi identity, making \mathfrak{g} into a Lie algebra over \mathbb{R}. This structure captures the infinitesimal symmetries of G, linearizing the nonlinear group operations near the identity. Equivalently, \mathfrak{g} may be constructed as the real vector space of all left-invariant vector fields on G, where a vector field X on G is left-invariant if it satisfies d(L_g)_h (X_h) = X_{gh} for all g, h \in G and the differential d(L_g)_h of the left translation L_g: k \mapsto gk. Under this identification, any X \in T_e G extends to a unique left-invariant vector field \tilde{X} on G by \tilde{X}_g = d(L_g)_e (X) for g \in G, providing a canonical isomorphism between T_e G and the space of left-invariant vector fields. The Lie bracket on \mathfrak{g} then coincides with the Lie bracket of vector fields: for left-invariant fields \tilde{X}, \tilde{Y} corresponding to X, Y \in \mathfrak{g} and a smooth function f: G \to \mathbb{R},

[\tilde{X}, \tilde{Y}] f = \tilde{X}(\tilde{Y} f) - \tilde{Y}(\tilde{X} f),

which is itself left-invariant and bilinear over \mathbb{R}. The adjoint representation of G on \mathfrak{g} arises naturally from the conjugation action on G: define \mathrm{Ad}: G \to \mathrm{GL}(\mathfrak{g}) by \mathrm{Ad}_g (X) = d(c_g)_e (X) for X \in \mathfrak{g}, where c_g: h \mapsto g h g^{-1} is conjugation by g and d(c_g)_e is its differential at e. Equivalently, \mathrm{Ad}_g (X) = d(L_g)_e \circ d(R_{g^{-1}})_e (X), with R_{g^{-1}} the right translation by g^{-1}; this yields a Lie group homomorphism whose differential at e is the adjoint representation \mathrm{ad}: \mathfrak{g} \to \mathrm{End}(\mathfrak{g}) of the Lie algebra on itself, given by \mathrm{ad}_X (Y) = [X, Y]. Another construction identifies elements of \mathfrak{g} with derivatives of one-parameter subgroups: a one-parameter subgroup of G is a smooth group homomorphism \gamma: (\mathbb{R}, +) \to G, and \mathfrak{g} consists of all such \gamma'(0) \in T_e G as \gamma varies, with the Lie bracket induced from the vector field structure. By Cartan's closed subgroup theorem, every closed subgroup H of G (in the subspace topology) is itself a Lie subgroup, hence a smooth submanifold of G with the induced smooth structure, and its Lie algebra \mathfrak{h} = T_e H forms a Lie subalgebra of \mathfrak{g}. This correspondence holds under the condition that H is closed, ensuring the submanifold structure and compatibility of the bracket.

Exponential Map and Properties

The exponential map serves as a fundamental connection between the Lie algebra \mathfrak{g} of a Lie group G and the group G itself, facilitating local approximations of the group structure near the identity. For X \in \mathfrak{g}, the exponential map \exp: \mathfrak{g} \to G is defined by \exp(X) = \gamma(1), where \gamma: \mathbb{R} \to G is the integral curve of the left-invariant vector field corresponding to X, satisfying the differential equation \gamma'(t) = dL_{\gamma(t)}(X) with initial condition \gamma(0) = e, and L_g denotes left multiplication by g \in G.^[32] This construction ensures that \exp captures one-parameter subgroups generated by elements of the Lie algebra.^[32] The exponential map possesses several key properties that underscore its role in Lie theory. It is a smooth map, with \exp(0) = e, and its differential at the origin d(\exp)_0: \mathfrak{g} \to T_e G is the identity isomorphism, identifying the Lie algebra with the tangent space at the identity.^[32] Near $0 \in \mathfrak{g}, \exp is a local diffeomorphism, allowing it to provide a neighborhood of the identity in G diffeomorphic to a neighborhood of $0 in \mathfrak{g}, which is essential for analyzing the local structure of Lie groups.^[32] For matrix Lie groups, where G is realized as a closed subgroup of \mathrm{GL}(n, \mathbb{C}) and \mathfrak{g} \subset \mathfrak{gl}(n, \mathbb{C}), the exponential map admits an explicit power series expression \exp(X) = \sum_{k=0}^\infty \frac{X^k}{k!}, which converges for all X \in \mathfrak{g} due to the analytic nature of matrix exponentials.^[56] A crucial tool for understanding group multiplication via the Lie algebra is the Baker-Campbell-Hausdorff (BCH) formula, which provides a formal power series expression for the logarithm of a product of exponentials. Specifically, \log(\exp(X) \exp(Y)) = X + Y + \frac{1}{2}[X, Y] + higher-order terms involving nested Lie brackets, where the series converges in a neighborhood of $0 and terminates exactly when the Lie algebra is nilpotent, yielding a polynomial relation that fully determines the group law locally from the algebra.^[57] This formula highlights how commutators in \mathfrak{g} encode the non-commutativity of G.^[57] While the exponential map is always surjective onto a neighborhood of the identity, it is generally not surjective onto all of [G](/page/G), reflecting global topological features. A classic illustration arises in the relationship between the rotation groups \mathrm{SO}(3) and its double cover \mathrm{SU}(2): the Lie algebras \mathfrak{so}(3) and \mathfrak{su}(2) are isomorphic, and the covering map \mathrm{SU}(2) \to \mathrm{SO}(3) induces a two-to-one correspondence such that each element of \mathrm{SO}(3) corresponds to two preimages under the composition of exponentials, demonstrating how the exponential map's fibers can reveal covering structures without the map itself being non-surjective for these groups.

Structural Properties

Homomorphisms and Isomorphisms

A Lie group homomorphism between two Lie groups G and H is a map \phi: G \to H that is both a smooth map of manifolds and a group homomorphism.^[56] Such a homomorphism induces a Lie algebra homomorphism d\phi_e: \mathfrak{g} \to \mathfrak{h} at the identity element e, given by the differential of \phi at e, which preserves the Lie bracket.^[32] For connected Lie groups, this induced map fully determines the homomorphism locally near the identity via the exponential map, as \phi(\exp_G(X)) = \exp_H(d\phi_e(X)) for X \in \mathfrak{g} sufficiently small, and global extension follows by connectivity.^[58] The kernel of a Lie group homomorphism \phi: G \to H is a closed normal Lie subgroup of G, and its Lie algebra is the kernel of d\phi_e.^[32] For covering maps between Lie groups, the kernel is discrete, consisting of a finite or countable central subgroup.^[51] The image \phi(G) is an immersed Lie subgroup of H, and \phi is an isomorphism if it is bijective and has a smooth inverse, ensuring both groups are diffeomorphic as manifolds.^[32] A prominent example is the inclusion SO(n) \hookrightarrow O(n), which is a smooth injective homomorphism with discrete kernel \{I\}, preserving the orthogonal group structure while restricting to positive determinant matrices.^[59] Another key case is the quotient map SU(2) \to SO(3) with kernel \{\pm I\}, yielding the isomorphism SO(3) \cong SU(2)/\{\pm I\}, which identifies the rotation group with the projective special unitary group.^[60] Automorphisms of a Lie group G are isomorphisms \phi: G \to G, forming the automorphism group \mathrm{Aut}(G). Inner automorphisms are those of the form \mathrm{Ad}_g(h) = g h g^{-1} for g \in G, comprising the normal subgroup \mathrm{Inn}(G) \cong G / Z(G), where Z(G) is the center.^[59] Outer automorphisms form the quotient \mathrm{Out}(G) = \mathrm{Aut}(G) / \mathrm{Inn}(G), and for semisimple Lie groups, they relate to the adjoint group, the image of the adjoint representation G \to \mathrm{Aut}(\mathfrak{g}), which is G modulo its center.^[61]

Subgroups and Simply Connected Groups

A Lie subgroup of a Lie group G is a subgroup H \subseteq G that is itself a Lie group under the subspace topology and the induced smooth structure, such that the inclusion map i: H \to G is a smooth Lie group homomorphism.^[62] This requires H to be an immersed submanifold of G, meaning i is an immersion (injective on tangent spaces), and the group operations restricted to H remain smooth.^[62] Immersed Lie subgroups need not be closed in the topological sense; for instance, the image of the embedding \mathbb{R} \to \mathbb{T}^2 given by t \mapsto (e^{2\pi i t}, e^{2\pi i \alpha t}), where \alpha is irrational, yields a dense immersed subgroup that is not closed.^[1] Closed subgroups of a Lie group G, by contrast, are embedded submanifolds and inherit a full Lie group structure automatically. The closed subgroup theorem asserts that every closed subgroup of a Lie group is itself a Lie group, resolving the question of when a topological subgroup admits a compatible smooth structure.^[63] This result follows from the solution to Hilbert's fifth problem, which proves that every locally Euclidean topological group is a Lie group; the resolution was independently achieved by Gleason and by Montgomery and Zippin in the early 1950s, showing that no additional axioms beyond local compactness and continuity are needed.^[64] Modern analytic criteria for immersed subgroups emphasize that they arise as images of smooth homomorphisms from other Lie groups, but density or non-closedness can prevent embedding unless the image is closed.^[62] Integral subgroups provide another perspective, generated by one-parameter subgroups via the exponential map. For a subalgebra \mathfrak{h} \subseteq \mathfrak{g}, the integral subgroup is the subgroup generated by elements of the form \exp(tX) for X \in \mathfrak{h}, t \in \mathbb{R}, forming an immersed Lie subgroup whose Lie algebra is \mathfrak{h}; this construction yields the connected component of the identity in the closed subgroup it generates.^[65] A connected Lie group G may not be simply connected, but it admits a unique simply connected covering Lie group \hat{G}, up to isomorphism over G, via a covering homomorphism \pi: \hat{G} \to G that is a local diffeomorphism with discrete kernel isomorphic to the fundamental group \pi_1(G).^[1] The universal cover \hat{G} is a Lie group, and \pi preserves the group structure, with central kernel.^[1] For example, the universal cover of the circle group S^1 is \mathbb{R} under the exponential map t \mapsto e^{2\pi i t}, with kernel \mathbb{Z}.^[1] In the semisimple case, the simply connected cover of a connected semisimple Lie group has finite center, reflecting the discreteness of the kernel in the covering.^[3]

Representations

Group Representations on Vector Spaces

A representation of a Lie group G on a finite-dimensional vector space V over \mathbb{R} or \mathbb{C} is a smooth group homomorphism \rho: G \to \mathrm{GL}(V), where \mathrm{GL}(V) denotes the general linear group of invertible linear transformations on V.^[66] Such representations encode the action of the group elements as linear maps, preserving the group structure through composition.^[67] For Lie groups, the smoothness condition ensures that the representation is compatible with the manifold structure of G.^[68] The associated Lie algebra representation is obtained by differentiating \rho at the identity element e \in [G](/page/G), yielding a Lie algebra homomorphism d\rho_e: \mathfrak{g} \to \mathfrak{gl}(V), where \mathfrak{g} is the Lie algebra of G and \mathfrak{gl}(V) is the Lie algebra of \mathrm{[GL](/page/GL)}(V).^[69] This differential satisfies the commutation relation [d\rho_e(X), d\rho_e(Y)] = d\rho_e([X, Y]) for all X, Y \in \mathfrak{g}, preserving the Lie bracket structure.^[70] Thus, every smooth representation of the Lie group induces a representation of its Lie algebra, providing a linear approximation near the identity.^[68] Unitary representations arise when V is equipped with an inner product and \rho(g) preserves this inner product for all g \in G, meaning \rho(g) is a unitary operator on the finite-dimensional Hilbert space V.^[71] For compact Lie groups, every finite-dimensional representation is equivalent to a unitary one, a consequence of the existence of a positive-definite invariant inner product.^[67] The Peter-Weyl theorem further asserts that the matrix coefficients of all finite-dimensional irreducible unitary representations of a compact Lie group form an orthonormal basis for the Hilbert space of square-integrable functions on G with respect to the Haar measure.^[71] A canonical example is the adjoint representation \mathrm{Ad}: G \to \mathrm{GL}(\mathfrak{g}), defined by \mathrm{Ad}_g(X) = g X g^{-1} for X \in \mathfrak{g}, which acts on the Lie algebra by conjugation and is smooth since conjugation is a diffeomorphism.^[72] For the special unitary group \mathrm{SU}(n), the fundamental representation is the standard n-dimensional action on \mathbb{C}^n, where group elements act by matrix multiplication, preserving the Hermitian inner product.^[73] This representation is irreducible and generates all others under tensor products and symmetries.^[74] In physics applications, the infinitesimal generators of a unitary representation are the operators i \, d\rho_e(X) for X \in \mathfrak{g}, which are self-adjoint with respect to the inner product on V and generate one-parameter subgroups via the exponential map.^[75] For matrix Lie groups, the group representation satisfies \rho(\exp(X)) = \exp(d\rho_e(X)) for X \in \mathfrak{g}.^[66] These generators underpin symmetry transformations in quantum mechanics, such as rotations and boosts.^[20]

Irreducible Representations and Characters

An irreducible representation of a Lie group G on a finite-dimensional complex vector space V is a representation \rho: G \to \mathrm{GL}(V) that admits no proper nontrivial invariant subspaces, meaning there exists no subspace W \subset V with $0 < \dim W < \dim V such that \rho(g)W = W for all g \in G.^[76] Schur's lemma states that if \rho is irreducible, then the only endomorphisms of V that commute with all \rho(g) form a division algebra over \mathbb{C}, which is isomorphic to \mathbb{C} for complex representations, implying that such commuting operators are scalar multiples of the identity.^[76] For compact Lie groups, every finite-dimensional representation decomposes into a direct sum of irreducible representations; this follows from Weyl's unitarity trick, which embeds the representation into a unitary one on L^2(G) via averaging with respect to the Haar measure, allowing the application of spectral theory to achieve complete reducibility.^[71] The characters of these representations provide a key tool for decomposition: the character \chi_\rho of an irreducible representation \rho is defined by \chi_\rho(g) = \mathrm{tr}(\rho(g)) for g \in G.^[77] The Schur orthogonality relations for compact groups assert that, with respect to the normalized Haar measure dg on G,

\int_G \chi_\rho(g) \overline{\chi_\sigma(g)} \, dg = \delta_{\rho\sigma} \frac{1}{\dim \rho},

where \delta_{\rho\sigma} is the Kronecker delta, confirming that distinct irreducibles are orthogonal and enabling the projection onto isotypic components. In the representation theory of semisimple Lie groups, highest weight theory classifies irreducible representations via dominant weights, which are linear functionals on a Cartan subalgebra \mathfrak{h} (a maximal abelian subalgebra of the Lie algebra \mathfrak{g}) that are nonnegative on the positive root system.^[78] Each irreducible representation is uniquely determined up to isomorphism by its highest weight, a dominant integral weight \lambda \in \mathfrak{h}^* such that the representation is generated by a highest weight vector v satisfying \rho(H)v = \lambda(H)v for H \in \mathfrak{h} and annihilated by positive root vectors.^[79] In physics, irreducible representations of Lie groups underpin the analysis of symmetry breaking, where the vacuum state transforms under a nontrivial irrep of the symmetry group, leading to spontaneous breaking of continuous symmetries and the emergence of Goldstone bosons in effective field theories.^[80] For instance, in the standard model, the Higgs mechanism involves irreps of the electroweak SU(2) × U(1) group to break symmetry while preserving gauge invariance.^[5]

Classification

Compact Lie Groups

Compact connected Lie groups admit a rich structure that allows for their complete classification. Every such group G decomposes as a finite quotient of a direct product of a torus and a simply connected semisimple compact Lie group. Specifically, G is isomorphic to (T \times \tilde{K})/D, where T is a torus, \tilde{K} is the simply connected cover of the semisimple part of G, and D is a finite discrete central subgroup.^[81] This decomposition arises because the center of a connected compact Lie group is contained in every maximal torus, and the semisimple quotient is obtained by factoring out the solvable radical. A key role in this structure is played by maximal tori. In a connected compact Lie group G, a maximal torus T is a closed connected abelian subgroup that is maximal with respect to inclusion, and all maximal tori are conjugate under G. The centralizer C_G(T) of T in G is T itself, ensuring that T is abelian and that the normalizer N_G(T) controls the symmetries.^[82] This setup facilitates the study of the group's representation theory, where weights lie in the weight lattice associated with the root system.^[83] The classification of connected compact Lie groups reduces to that of their Lie algebras, which are reductive. The semisimple part is classified by the Killing-Cartan theorem, which enumerates the finite-dimensional simple complex Lie algebras up to isomorphism via their root systems. These are labeled by Dynkin diagrams of types A_n, B_n, C_n, D_n, E_6, E_7, E_8, F_4, and G_2. For the compact real forms, type A_n corresponds to the Lie algebra \mathfrak{su}(n+1) of the special unitary group SU(n+1), B_n to \mathfrak{so}(2n+1) of SO(2n+1), C_n to \mathfrak{sp}(n) of the compact symplectic group [Sp](/page/SP)(n), and D_n to \mathfrak{so}(2n) of SO(2n), with exceptional types yielding the corresponding compact exceptional groups.^[84]^[85] Associated to the root system is the Weyl group W, defined for a maximal torus T as W = N_G(T)/T, which is a finite reflection group acting faithfully on the dual space \mathfrak{t}^* of the Lie algebra of T. This action is generated by reflections s_\alpha across the hyperplanes perpendicular to the roots \alpha, preserving the root lattice and inducing symmetries on the weight lattice. The ring of polynomial invariants under W is freely generated by r homogeneous polynomials, known as the fundamental invariants, where r is the rank of the root system; their degrees are determined by the structure of the Dynkin diagram.^[86] A fundamental existence theorem states that every connected compact Lie group G is isomorphic to the quotient of its simply connected universal cover \tilde{G} by a discrete central subgroup \Gamma \cong \pi_1(G), which is finite since \tilde{G} is compact. For semisimple G, \tilde{G} is also semisimple and compact. Representative examples include SU(2), which is simply connected with fundamental group trivial, and SO(3), which is the quotient SU(2)/\mathbb{Z}_2 where \mathbb{Z}_2 = \{\pm I\} is the center of SU(2).^[87]^[88]

Semisimple and Nilpotent Cases

A semisimple Lie algebra over a field of characteristic zero is defined as one whose radical is zero, meaning it has no nonzero solvable ideals. Such algebras admit a decomposition as a direct sum of simple Lie algebras, where simple Lie algebras are those with no nontrivial ideals. This direct sum structure implies that the adjoint representation of a semisimple Lie algebra is completely reducible, reflecting the absence of solvable components.^[89] For general finite-dimensional Lie algebras over fields of characteristic zero, the Levi decomposition theorem provides a canonical splitting: any such Lie algebra \mathfrak{g} is isomorphic to a semidirect product \mathfrak{g} = \mathfrak{s} \ltimes \mathfrak{r}, where \mathfrak{s} is a semisimple subalgebra (the Levi factor) and \mathfrak{r} is the radical of \mathfrak{g} (its maximal solvable ideal). This decomposition highlights how semisimple components interact with solvable ones via actions on the radical. An example of a semisimple Lie algebra that is non-compact is \mathfrak{sl}(2, \mathbb{R}), the Lie algebra of $2 \times 2 real matrices with trace zero, which is simple and admits a non-degenerate but indefinite Killing form.^[90]^[32] Solvable Lie algebras form a broader class, characterized by the termination of their derived series: define \mathfrak{g}^{(0)} = \mathfrak{g} and \mathfrak{g}^{(k+1)} = [\mathfrak{g}^{(k)}, \mathfrak{g}^{(k)}] for k \geq 0, so \mathfrak{g} is solvable if \mathfrak{g}^{(k)} = 0 for some k. Lie's theorem establishes a key representation-theoretic property: over an algebraically closed field of characteristic zero, every finite-dimensional representation of a solvable Lie algebra on a vector space admits a basis in which the representing matrices are upper triangular. This triangularizability implies that the eigenvalues of the representation lie in the base field and underscores the "unipotent-like" behavior of solvable algebras. A representative example is the Lie algebra of the affine group \mathrm{Aff}(\mathbb{R}), consisting of transformations x \mapsto ax + b with a \neq 0, whose Lie algebra is the semidirect product \mathbb{R} \ltimes \mathbb{R} (with the first factor acting by scaling); its derived algebra is the translation subalgebra \mathbb{R}, which is abelian, so the derived series terminates after two steps.^[91]^[92]^[93] Nilpotent Lie algebras are a special subclass of solvable ones, defined by the termination of the lower central series: set \mathfrak{g}_0 = \mathfrak{g} and \mathfrak{g}_{k+1} = [\mathfrak{g}, \mathfrak{g}_k] for k \geq 0, so \mathfrak{g} is nilpotent if \mathfrak{g}_k = 0 for some k. Engel's theorem provides an equivalent condition: a Lie algebra is nilpotent if and only if the adjoint operator \mathrm{ad}_x is nilpotent for every x \in \mathfrak{g}. For simply connected nilpotent Lie groups, the exponential map \exp: \mathfrak{g} \to G is a global analytic diffeomorphism, allowing a direct correspondence between the algebra and group structures. The Heisenberg group, realized as upper triangular $3 \times 3 matrices with ones on the diagonal, exemplifies this: its Lie algebra has lower central series terminating at step 2, and the exponential map identifies the group with \mathbb{R}^3 equipped with a non-commutative multiplication.^[94]^[94]^[95]

Infinite-Dimensional Extensions

General Framework

Infinite-dimensional Lie groups extend the classical theory of finite-dimensional Lie groups to settings where the underlying manifold has infinite dimension, presenting significant challenges due to the lack of a complete finite-dimensional analogy. In finite dimensions, Lie groups are smooth manifolds with compatible group structure, but in infinite dimensions, the absence of a canonical notion of smoothness necessitates modeling on topological vector spaces (TVS), such as Fréchet or Banach spaces, to define differentiable structures. This framework addresses Hilbert's fifth problem, which characterizes topological groups locally Euclidean as Lie groups in the finite-dimensional case, but reveals pathologies in infinite dimensions where not all continuous groups admit Lie group structures without additional topological assumptions. A key prerequisite for infinite-dimensional Lie theory is familiarity with functional analysis, particularly locally convex topological vector spaces, which provide the model spaces for charts in the manifold structure. Infinite-dimensional Lie groups are typically defined as groups that are smooth manifolds modeled on such TVS, where the group operations—multiplication and inversion—are smooth maps in the sense of the manifold's differential structure. For Fréchet Lie groups, the model space is a Fréchet space, allowing for countable decreasing sequences of seminorms to define the topology, while Banach Lie groups use complete normed spaces, though the former offers more flexibility for applications like diffeomorphism groups. Smoothness in this context requires that transition maps and group operations are Fréchet differentiable, extending the finite-dimensional exponential map from the Lie algebra to the group, though this map often lacks the same properties.^[96]^[97] Topological vector Lie groups form a broader class, consisting of Lie groups equipped with a topology making the vector space operations continuous and compatible with the group structure, often without requiring full smoothness. These groups are modeled directly on TVS, where the Lie algebra is also a TVS, and the adjoint representation is continuous. This setup contrasts with finite-dimensional cases by allowing for non-complete or non-metrizable topologies, which are essential for handling symmetries in infinite-dimensional settings like partial differential equations. However, the theory demands careful treatment of continuity to avoid counterexamples where topological groups fail to be Lie groups.^[98]^[99] The exponential map in infinite-dimensional Lie groups, which in finite dimensions provides a local diffeomorphism from the Lie algebra to the group via one-parameter subgroups, faces substantial obstacles. It may neither be surjective onto a neighborhood of the identity nor analytic, due to the potential failure of local solvability of ordinary differential equations in infinite-dimensional manifolds. Moreover, the Baker-Campbell-Hausdorff (BCH) formula, which formalizes the Lie algebra structure through logarithms of products in finite dimensions, often diverges in infinite dimensions because the series involves nested commutators that do not converge in the TVS topology. This divergence complicates the identification of the group with its Lie algebra near the identity, requiring alternative approaches like invariant metrics or semigroup approximations.^[100]^[101]^[102] Homomorphisms between infinite-dimensional Lie groups must preserve not only the algebraic structure but also the topological and smooth aspects, with continuity being indispensable for well-behaved representations. Unlike in finite dimensions, where the closed subgroup theorem guarantees that closed subgroups are Lie subgroups, this fails in infinite dimensions without additional conditions on the topology, leading to potential dense subgroups that are not closed. Thus, the study of homomorphisms often restricts to continuous or smooth maps between modeled spaces, ensuring compatibility with the differential structure.^[103]^[98]

Notable Examples and Applications

The diffeomorphism group \operatorname{Diff}(M) of a smooth manifold M consists of all smooth diffeomorphisms of M onto itself and forms an infinite-dimensional Lie group under composition, with its Lie algebra given by the space of smooth vector fields on M.^[104] For the specific case where M = S^1, the circle, \operatorname{Diff}(S^1) admits a unique nontrivial central extension known as the Virasoro group, whose Lie algebra is the Virasoro algebra—a central extension of the Witt algebra of vector fields on S^1—central to conformal field theory and two-dimensional gravity.^[105] This extension arises from a continuous cohomology class on the Lie algebra level, ensuring the group's projectivity and enabling quantization in physical models.^[106] Loop groups provide another fundamental class of infinite-dimensional Lie groups, defined as the group of smooth maps from the circle S^1 to a finite-dimensional Lie group G, equipped with the C^\infty topology and pointwise multiplication.^[107] These groups possess central extensions whose Lie algebras are affine Kac-Moody algebras, generalizing finite-dimensional semisimple Lie algebras through loop constructions and playing a pivotal role in integrable systems and conformal field theories.^[108] In gauge theories, loop groups model the symmetries of fields on spatial circles, facilitating the study of anomalies and topological invariants via their representations.^[109] The unitary group U(\mathcal{H}) on a separable infinite-dimensional Hilbert space \mathcal{H} comprises all unitary operators on \mathcal{H} and can be endowed with a Lie group structure using the strong operator topology, though analyticity requires careful choice of topology such as the norm or strong operator one.^[110] Its representations are intimately linked to C*-algebras, where irreducible unitary representations of U(\mathcal{H}) correspond to pure states in the C*-algebra of bounded operators on \mathcal{H}, enabling the spectral analysis of self-adjoint extensions and quantum mechanical observables.^[111] In applications, the diffeomorphism group \operatorname{Diff}(M) underlies the geometric formulation of ideal fluid dynamics, where the Euler equations describe geodesics on \operatorname{Diff}(M) equipped with a right-invariant L^2-metric, revealing stability and vorticity transport via group cohomology.^[112] Loop groups and their Kac-Moody extensions are essential in string theory, where they encode the symmetries of the closed string worldsheet, with central charges determining anomaly cancellation and modular invariance in critical dimensions.^[113] In quantum field theory, current algebras—modeled as infinite-dimensional Lie algebras of loop groups—govern the commutation relations of conserved currents, underpinning chiral symmetries and the operator product expansions in two-dimensional models.^[114]

References

[1]
[PDF] Introduction to Lie Groups and Lie Algebras Alexander Kirillov, Jr.
Lie Groups: Basic Definitions. Note that the definition of a Lie group does not require that G be connected. Thus, any finite group is a 0-dimensional Lie group ...
[2]
https://math.uchicago.edu/~womp/2007/lie2007.pdf
[3]
[PDF] Lie Groups. Representation Theory and Symmetric Spaces
Definition 1.1 A Lie group G is an abstract group and a smooth n- dimensional manifold so that multiplication G × G → G: (a, b) → ab and inverse G → G: a → a−1 ...
[4]
[PDF] historical overview of lie theory - Columbia Math Department
Jan 30, 2025 · We give a brief historical overview of Lie theory beginning with the work of Lie. Along the way, we introduce the fundamental definitions of Lie ...
[5]
[PDF] lie groups, lie algebras, and applications in physics - UChicago Math
Sep 17, 2015 · This paper introduces basic concepts from representation theory,. Lie group, Lie algebra, and topology and their applications in physics, par-.
[6]
Lie Groups and their applications to Particle Physics
We start from basic group theory and representation theory. We then introduce Lie Groups and Lie Algebra and their properties. We next discuss with detail two ...
[7]
[PDF] overview of lie group variational integrators and their applications to ...
Lastly, we demonstrate the application of Lie group variational integrators to the construction of optimal control algorithms on Lie groups, and describe a ...
[8]
Henri Poincaré - Stanford Encyclopedia of Philosophy
Sep 3, 2013 · Henri Poincaré was a mathematician, theoretical physicist and a philosopher of science famous for discoveries in several fields and referred to as the last ...
[9]
[PDF] The Evolution of Group Theory: A Brief Survey - Israel Kleiner
Mar 14, 2004 · Lie realized that the theory of continuous trans- formation groups was a very powerful tool in geometry and differential equations and he set.
[10]
A Century of Lie Theory - ROGER HOWE
In the case of Lie theory,. Sophus Lie was already studying “continuous, finite groups of transforma- tions" in the 1870s, and one could even make a case for ...
[11]
Felix Klein, Sophus Lie, contact transformations, and connexes.
As early as 1870, Lie studied particular examples of what he later called contact transformations, which preserve tangency and which came to play a crucial role ...Missing: 1873 Poincaré Cartan 1890s
[12]
Élie Cartan (1869 - 1951) - Biography - MacTutor
Cartan's doctoral thesis of 1894 contains a major contribution to Lie algebras where he completed the classification of the semisimple algebras over the ...Missing: date | Show results with:date
[13]
Hermann Weyl and representation theory | Resonance
Dec 17, 2016 · In 1925–26, Weyl wrote four epochal papers in representation theory of Lie groups which solved fundamental problems, and also gave birth to ...
[14]
[PDF] Hermann Weyl and Representation Theory
In 1925–26, Weyl wrote four path-breaking papers in representa- tion theory which apart from solving fundamental problems, also gave birth to the subject of ...
[15]
https://press.princeton.edu/books/paperback/9780691049908/theory-of-lie-groups
[16]
Essays in the history of Lie groups and algebraic groups, by Armand ...
Feb 12, 2003 · During the 1940s, the theory of algebraic groups attracted the attention first of. Chevalley and then of Kolchin. Motivated by his interest ...
[17]
[PDF] what does a lie algebra know about a lie group? - UChicago Math
We define Lie groups and Lie algebras and show how invariant vector fields ... Intuitively, Lie groups are smoothly-varying collections of symmetries.
[18]
[PDF] Lecture 4: Lie Groups - VNAV
Intuitively this means that these objects have nice group operations and admit a local parametrization. 4.2 Lie algebras. Every matrix Lie group is associated ...
[19]
Introduction to Lie Theory and its Application to Robotics
Lie theory is concerned with the study of Lie groups which are high-dimensional smooth manifolds with a group topology. The tangent space of these groups is a ...
[20]
[PDF] Lie Groups in Modern Physics - Oregon State University
May 14, 1996 · Lie groups were invented by the Swedish mathematician,. Sophus Lie in the late nineteenth century, but most of the theory that we use was ...Missing: early | Show results with:early
[21]
[PDF] Sophus Lie and the Role of Lie Groups in Mathematics By Sigurdur ...
The original founder of this theory was a Norwegian, Marius Sophus Lie, who was born in Nordfjordeid, 1842. In order to understand the background and motivation ...
[22]
[PDF] Introduction to Lie Algebras - UCI Mathematics
Lie theory has its roots in the work of Sophus Lie, who studied certain trans- formation groups that are now called Lie groups. His work led to the ...Missing: origins | Show results with:origins<|control11|><|separator|>
[23]
[PDF] Emmy Noether and Symmetry
laws of their corresponding field equations, one needs to gen- eralize Lie's concept of continuous symmetry group to include higher order generalized symmetries ...
[24]
[PDF] Noether's Theorems and Gauge Symmetries - arXiv
One theorem applies to symmetries associated with finite dimensional. Lie groups (global symmetries), and the other to symmetries associated with infinite ...
[25]
[PDF] Differential Geometry and Lie Groups A Second Course
Aug 14, 2025 · This book is written for a wide audience ranging from upper undergraduate to advanced graduate students in mathematics, physics, ...
[26]
[PDF] Hilbert Space Quantum Mechanics
In quantum mechanics unitary operators are used to change from one orthonormal basis to another, to represent symmetries, such as rotational symmetry, and ...
[27]
[2406.12571] The significance of the configuration space Lie group ...
Jun 18, 2024 · The proper c-space of a rigid body is the Lie group SE(3), and the geometry is that of the screw motions. The rigid bodies within a MBS are ...
[28]
[PDF] Applications of Lie Groups and Lie Algebra to Computer Vision
Most of literature about the applications of Lie groups in computer vision have focused on Affine group GA(2) [1, 2],. Rotation group SO(3) [3, 4, 5, 6], ...
[29]
[PDF] Deep Learning on Lie Groups for Skeleton-Based Action Recognition
The network structure is dubbed as LieNet, where each input is an element on the. Lie Group. Like convolutional networks (ConvNets), the. LieNet also exhibits ...
[30]
The symmetry groups of non-rigid molecules: a Lie algebraic and ...
Dec 3, 2010 · The description of the symmetry of non-rigid molecules is explored through a Lie algebraic approach. It is shown that the abstract process ...
[31]
Lie Group Modelling for an EKF-Based Monocular SLAM Algorithm
This paper addresses the problem of monocular Simultaneous Localization And Mapping on Lie groups using fiducial patterns.
[32]
[PDF] 18.745: lie groups and lie algebras, i - MIT Mathematics
Lie(B+) = b+ := h ⊕ n+; these are all closed Lie subgroups. Definition 49.1. A Borel subalgebra of g is a Lie subalgebra con- jugate to b+. A Borel ...
[33]
[PDF] Lie Groups: Fall, 2022 Lecture I - Columbia Math Department
Aug 24, 2022 · A Lie group is a smooth finite dimensional manifold G with two structure maps, which are required to be smooth maps, m: G×G → G and ι: G → G, ...Missing: mathematics | Show results with:mathematics
[34]
[PDF] Correspondence between Lie groups and Lie algebras.
Apr 16, 2023 · Manifolds. A space M is a topological manifold if it is Hausdorff, second countable and locally Euclidean, where locally Euclidean means ...
[35]
[PDF] Math 396. Paracompactness and local compactness
Any second countable Hausdorff space X that is locally compact is paracompact. Proof. Let {Vn} be a countable base of opens in X. Let {Ui} be an open cover ...
[36]
[PDF] 18.745 F20 Lecture 01: Manifolds - MIT OpenCourseWare
Also a subgroup of a topological group is itself a topological group, so another example is rational numbers with addition, (Q,+). This last example is not.
[37]
[PDF] Classification of Semisimple Lie Algebras
Definition 1. A Lie group is a smooth manifold G endowed with a group structure, such that the group operation and the inverse map are smooth.
[38]
[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 ...
[39]
[PDF] NOTES ON LIE GROUPS AND LIE ALGEBRAS (261) - UC Davis Math
There is Cartan's theorem which state that every subgroup of a Lie group G which is closed (in the topology of G) is a Lie subgroup of G; in particular, the ...
[40]
Sophus Lie (1842 - 1899) - Biography - MacTutor
It was during the winter of 1873-74 that Lie began to develop systematically what became his theory of continuous transformation groups, later called Lie groups ...Missing: smooth generalization
[41]
general linear group - PlanetMath.org
Mar 22, 2013 · The general linear group is an algebraic group, and it is a Lie group if V V is a real or complex vector space. When V V is a finite-dimensional ...
[42]
[PDF] Introduction to Lie groups, isometric and adjoint actions and ... - arXiv
We begin with GL(n, R), the general linear group of non singular2 n × n real matrices. Similarly, GL(n, C) and GL(n, H) are the groups of non singular ...
[43]
[PDF] Matrix Lie groups and their Lie algebras - Alen Alexanderian
Jul 12, 2013 · We discuss matrix Lie groups and their corresponding Lie algebras. Some common examples are provided for purpose of illustration. 1 Introduction.
[44]
Orthogonal Group -- from Wolfram MathWorld
For every dimension n>0, the orthogonal group O(n) is the group of n×n orthogonal matrices. These matrices form a group because they are closed under ...
[45]
dimension of the special orthogonal group - PlanetMath.org
Mar 22, 2013 · First observe that the set of orthogonal operators O(n) O ⁢ ( n ) is a manifold embedded in the real vector space GL(V)≃Rn×n GL ⁢ ( V ) ≃ ℝ ...
[46]
[PDF] Matrix Lie Groups and the Lie Group–Lie Algebra correspondence
Notice that SO is in fact a subgroup of O, as it is closed under multiplication and inversion and is a matrix Lie group for the same reasons as above.
[47]
unitary group in nLab
May 10, 2025 · The unitary group U ( n ) U(n) is compact topological space, hence in particular a compact Lie group. Homotopy groups. Proposition 2.2. For n , ...
[48]
[PDF] LIE GROUPS AND LIE ALGEBRAS - joseph redeker
SO(n) := {A ∈ O(n) | det(A)=1} Definition 2.8. The special unitary group is denoted SU(n) and defined as, SU(n) := {A ∈ U(n) | det(A)=1} We mention isometries ...
[49]
finite rotation group in nLab
Jul 22, 2025 · By a finite rotation group one means a finite subgroup of a group of rotations, hence of a special orthogonal group SO ( n ) SO(n) or spin group Spin ( n ) ...
[50]
[PDF] classifying the finite subgroups of so3 - UChicago Math
Aug 29, 2020 · In this paper, we classify the finite subgroups of SO3, the group of rotations of R3. We prove that all finite subgroups of SO3 are isomorphic ...
[51]
[PDF] Lecture Notes in Lie Groups - arXiv
Apr 7, 2011 · 4.6 Actions of Lie Groups on Smooth Manifolds . ... These continuous groups, which originally appeared as symmetry groups of dif-.
[52]
[PDF] Group Theory Essentials
Group multiplication then corresponds to the map which carries the pair θ, θ′ to θ + θ′ modulo 2π, whereas the inverse corresponds to θ → −θ. Θ Since SO(2) ...
[53]
[PDF] lie groups and lie algebras womp 2007 - UChicago Math
Definition. A Lie group is a group G that is also a smooth manifold, such that the multipli- cation G × G → G, (g ...Missing: mathematics | Show results with:mathematics
[54]
[PDF] Lie Groups - UC Berkeley math
May 25, 2012 · A typical example of a solvable Lie group is the group of upper triangular matrices with nonzero determinant. (Recall that solvable means the ...
[55]
[PDF] arXiv:2311.17899v2 [math.DG] 20 Feb 2024
Feb 20, 2024 · • The abelian Lie group (R3, +);. • The 3-dimensional real Heisenberg group H3(R), that is the group of upper uni- triangular 3-by-3 real ...
[56]
[PDF] Introduction to representation theory - MIT Mathematics
Jan 10, 2011 · The Heisenberg Lie algebra H of matrices. 0 ∗ ∗. 0 0 ∗. 000. It has the basis x =. 0 0 0. 0 0 1. 0 0 0.. y =. 0 1 0. 0 0 0. 0 0 0.
[57]
[PDF] 18.199 Talk 1 : A Crash Course on Lie Groups
Definition 1.1. A set G is a Lie group if it is a group and a smooth manifold such that multipli- cation and inversion maps are smooth.
[58]
[PDF] Prove the Baker–Campbell–Hausdorff formula - MIT Mathematics
Dec 6, 2017 · Two of the main applications of the Baker–Campbell–Hausdorff formula is its role in proving the Lie group–Lie algebra correspondence, which ...
[59]
[PDF] Lie Groups - U.C. Berkeley Mathematics
Oct 5, 2016 · 1.1.2 Analytic and algebraic groups. 1.1.2.1 Definition A Lie group is a group object in a category of manifolds. In particular, a. Lie group ...
[60]
[PDF] Lie groups and Lie algebras (Winter 2024)
Lie group actions. Definition 7.1. An action of a Lie group G on a manifold M is a group homomor- phism. A: G → Diff(M), g 7→ Ag into the group of ...
[61]
[PDF] Topology of SO(3) for Kids arXiv:2310.19665v1 [math.HO] 30 Oct 2023
Oct 30, 2023 · In other words, π1(SO(3)) is isomorphic to the group of two elements ... SU(2) → SO(3), which is a local isomorphism, and which sends ...
[62]
[PDF] Actions of automorphism groups of Lie groups - arXiv
Mar 28, 2017 · The automorphism group of a connected Lie group can be realised as a linear group via association with the corresponding automorphism of the Lie.
[63]
[PDF] lecture 18: lie subgroups
Theorem 2.3 (E. Cartan's closed subgroup theorem). Any closed subgroup H of a Lie group G is a Lie subgroup (and thus a smooth submanifold) of G. Remark.
[64]
[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.
[65]
[PDF] 2015.84624.Topological-Transformation-Groups.pdf
Jul 14, 1970 · TOPOLOGICAL TRANSFORMATION GROUPS. Deane Montgomery and Leo Zippin. INTERSCIENCE PUBLISHERS, INC., NEW YORK. INTERSCIENCE PUBLISHERS LTD ...
[66]
[PDF] 8. The Lie algebra and the exponential map for general Lie groups
Moreover, using the exponential map obtained from the left invariant theory we can obtain the right invariant vector fields and their integral curves very ...
[67]
[PDF] Simply Connected Lie Groups & the Exponential Map
A simply connected Lie group is path-connected and its fundamental group is trivial. If Lie groups have isomorphic Lie algebras, they are not necessarily ...
[68]
[PDF] 18.757 F23 Full Lecture Notes: Representations Of Lie Groups
Introduction. These notes are based on the course “Representations of Lie groups” taught by the author at MIT in Fall 2021 and Fall 2023. This is the.
[69]
5.1 Basics‣ 5 Representations of Lie groups and Lie algebras
A finite-dimensional (complex) representation ( ρ , V ) of a Lie group G is a Lie group homomorphism where V is a finite-dimensional complex vector space.
[70]
[PDF] representations of lie groups and lie algebras
Oct 1, 2013 · In this section, we shall introduce the notion of Lie group and its Lie algebra. Since a Lie group is a smooth manifold, we shall also introduce ...
[71]
[PDF] Background on representations of Lie groups and Lie algebras
Definition 2 (Lie algebra Representation). A representation of a Lie algebra g is a module for the algebra U(g). This module is given by a choice of vector.
[72]
[PDF] BASIC REPRESENTATION THEORY 1. Representations of Lie ...
A representation of a Lie group G is a pair (V,π), where V is a vector space, and π : G → GL(V ) is a linear G-action on V . Remarks. (1) Although a ...
[73]
[PDF] Representations of Compact Lie Groups - MIT OpenCourseWare
A fundamental result about compact Lie groups is that this system is, in fact, complete: Theorem 35.5. (Peter-Weyl theorem) The functions ψV,ij form an.
[74]
5.3 The adjoint representation
Let G be a linear Lie group and g be its Lie algebra. Then the adjoint representation Ad of G is the action on g by conjugation.
[75]
[PDF] Week 14: Group theory primer 1 SU(N) - UCSB Physics
A representation of the Lie algebra is a realization of these commutation relations on a set of M × M matrices. The theory of the group SU(N) begins with a ...
[76]
[PDF] 2A. SU(n), SO(n), and Sp(2n) Lie groups * version 1.3 *
Sep 12, 2016 · Here we define unitary, orthogonal, and symplectic Lie groups via their fundamental representations. This is a brief “first pass” to.
[77]
[PDF] Chapter 1 LIE GROUPS - INFN
Jan 1, 2011 · Thus, the group structure of a Lie group is etirely determined, via the exponential map, from the Lie algebra of its infinitesimal generators.
[78]
[PDF] Introduction to representation theory - arXiv
Feb 1, 2011 · The notes cover a number of standard topics in representation theory of groups, Lie algebras, and ... By the usual Schur's lemma, the algebra D := ...
[79]
[PDF] Math 210B. Characters for compact Lie groups
In particular, all of the “classical matrix groups” that are closed subgroups of GLn(R) or GLn(C) are Lie groups, such as: SLn(R), SLn(C),. Sp2g(R), Sp2g(C), ...
[80]
[PDF] Representations of Semisimple Lie Groups
Lecture 4 specializes to the representation theory of compact connected Lie groups, where the Theorem of the Highest Weight parametrizes the irreducible ...
[81]
[PDF] REPRESENTATION OF SEMISIMPLE LIE ALGEBRAS
(1) Every Irreducible representation has a highest weight. (2) Two irreducible representations with the same weight are equivalent. (3) The highest weight of ...
[82]
[PDF] Groups and Representations
We can define reducible, irreducible and fully reducible representations of Lie algebras in ... Application 5.6: (Spontaneous breaking of (global) Lie symmetries).
[83]
[PDF] math 210c. compact lie groups - Harvard Mathematics Department
Functoriality of brackets. Definition 4.10. For any given Lie group, we let the lower-case fraktur font denote the corresponding Lie algebra. So for example ...
[84]
https://math.uchicago.edu/~may/REU2012/REUPapers/Bosshardt.pdf
[85]
[PDF] The Killing Form, Reflections and Classification of Root Systems 1 ...
One can show that for semi-simple Lie algebras the Killing form is non-degenerate. For semi-simple Lie groups the Killing form can be used to define an inner.
[86]
[PDF] the classification of simple complex lie algebras - UChicago Math
Aug 24, 2012 · We have thus reduced the project of classifying simple Lie algebras to determining connected Dynkin diagrams, which we can cat- alog by a ...Missing: compact | Show results with:compact
[87]
[PDF] PART II: Classification of semi-simple Lie algebras. - DAMTP
Given a Cartan matrix, one can reconstruct the simple roots of the algebra. Definition 8. The Dynkin diagram associated with a complex semi-simple Lie algebra ...
[88]
[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 ...
[89]
[PDF] 18 Root systems and reflection groups
Most but not all reflection groups in Euclidean space turn up as Weyl groups of Lie groups: the exceptions are most dihedral groups, the symmetries of an ...Missing: compact | Show results with:compact
[90]
[PDF] Lecture Notes on Compact Lie Groups and Their Representations
3.2.15 Theorem (Weyl) Let G be a compact connected semisimple Lie group. Then the universal covering Lie group. ˜. G is also compact (equivalently, the fun-.
[91]
[PDF] lie theory and topology - allen knutson
SU(2)/Z2 ∼= SO(3). That gives three obvious quotients of SU(2) × SU(2). The ... connected compact Lie group K. Then the map. [V] 7→ arg max λ∈T*. hX ...
[92]
[PDF] Lecture 11 — The Radical and Semisimple Lie Algebras
Oct 14, 2010 · Any semisimple, finite-dimensional Lie algebra over a field F of charac- teristic 0 is a direct sum of simple Lie algebras. Proof. If g is ...
[93]
[PDF] Lecture 23 — Decomposition of Semisimple Lie Algebras
Dec 2, 2010 · This will be used to prove the Weyl's Complete Reducibility Theorem and Levi's Theorem. The following exercise follows from the definitions and ...Missing: simple | Show results with:simple
[94]
[PDF] 18.757 (Representation of Lie Algebras) Lecture Notes - Evan Chen
May 12, 2016 · The Lie algebra g is solvable if the derived series. D0g = g,. Dig = [Di−1g,Di−1g] eventually terminates. Definition 7.4. The Lie algebra g is ...
[95]
[PDF] Lecture 5 — Lie's Theorem
Sep 23, 2010 · This concludes the proof. Lie's Theorem. Let g be a solvable Lie algebra and π a representation of g on a finite dimensional vector space V ...
[96]
[PDF] Lie Algebras, Algebraic Groups, and Lie Groups - James Milne
May 5, 2013 · . PROPOSITION 3.2 A Lie algebra g is solvable if and only if its derived series terminates with zero. PROOF. If the derived series ...
[97]
[PDF] Nilpotent Lie Algebras and Engel's Theorem
The descending central series for g is the sequence of ideals g = c1(g) ⊃ c2(g) ⊃ ··· ⊃ ci(g) ⊃ ··· . (Since dim g is finite, it is clear that this series ...Missing: lower exponential
[98]
[PDF] arXiv:2011.13665v1 [math.GR] 27 Nov 2020
Nov 27, 2020 · It is well known that the exponential map exp: g → G is a global analytic diffeomorphism whenever G is a simply connected nilpotent Lie group,.<|separator|>
[99]
https://www.math.uni-hamburg.de/home/wockel/data/monastir.pdf
[100]
[PDF] The Geometry of Infinite-Dimensional Groups - ReadingSample - NET
In particular, our infinite-dimensional Lie groups are Fréchet Lie groups. Instead of Fréchet manifolds, one could consider manifolds modeled on Ba- nach spaces ...
[101]
[PDF] Infinite-Dimensional Lie Groups - HAL
Jun 4, 2009 · The differential-geometric approach to finite-dimensional global Lie groups (as smooth or analytic manifolds) is naturally complemented by the ...
[102]
[PDF] Monastir Summer School: Infinite-Dimensional Lie Groups
Jan 9, 2006 · In the introductory section, we present some of the main types of infinite-dimensional Lie groups: linear Lie groups, groups of ... Karl-Hermann ...
[103]
[PDF] Fundamental Problems in the Theory of Infinite-Dimensional Lie ...
Feb 5, 2006 · Let M be a compact manifold, G a Lie group with a smooth exponential map expG (e.g., a finite-dimensional Lie group). Then. C∞(M,G) is a group ...Missing: formula | Show results with:formula
[104]
[PDF] HILBERT-SCHMIDT GROUPS AS INFINITE-DIMENSIONAL LIE ...
We describe the exponential map from an infinite-dimensional Lie algebra to an infinite-dimensional group of operators on a Hilbert space. No- tions of ...
[105]
[PDF] Towards a BCH Formula on the Diffeomorphism Group with a Right ...
Dec 22, 2023 · To date, there is no BCH formula for the diffeomorphism groups, although they are infinite dimensional analogues of Lie groups, because the ...
[106]
Infinite-Dimensional Lie Groups - AMS Bookstore
This book develops, from the viewpoint of abstract group theory, a general theory of infinite-dimensional Lie groups involving the implicit function theorem ...<|separator|>
[107]
[PDF] Finite and Infinite Dimensional Lie Groups And Evolution Equations ...
The Virasoro group and KdV. The group Diff(S1) has a central extension called the Virasoro group, whose Lie algebra. (9.1). Vir(S1) = Vect(S1) ⊕ R ≈ C∞(S1) ...
[108]
[PDF] The Virasoro algebra and its representations in physics
2 The Virasoro algebra as a central extension. The Virasoro algebra is actually the unique central extension of the Lie algebra of the group Diff(S1) of ...
[109]
[PDF] Central extensions of infinite-dimensional Lie groups - Numdam
infinite-dimensional Lie groups have such open covers. It would be very ... with the other one for Frechet manifolds, i.e., manifolds modeled on Frechet ...
[110]
[PDF] Loop groups
One reason for study- ing such groups is that they are the simplest examples of infinite dimensional Lie groups. Thus LG has a Lie algebra L~ - the loops in the ...
[111]
[PDF] infinite dimensional lie groups and applications
Course description. Kac-Moody groups are natural generalizations to infinite dimensions of finite dimensional simple Lie groups. The subclass of real forms ...
[112]
[PDF] arXiv:1005.0495v2 [hep-th] 10 Jul 2010
Jul 10, 2010 · The most immediate generalization of the loop group algebra (or Kac-Moody algebra) is the algebra corresponding to the Lie group Map(M; G) of ...<|control11|><|separator|>
[113]
Infinite-Dimensional Groups and Their Representations - SpringerLink
This article provides an introduction to the representation theory of Banach-Lie groups of operators on Hilbert spaces.
[114]
[PDF] Infinite Dimensional Lie Theory
Briefly, it is a C∗-algebra L whose multiplier algebra M(L) admits a homomorphism η : G → U(M(L)), such that the (unique) extension of the representation theory.
[115]
[PDF] Diffeomorphism groups, hydrodynamics and - relativity
By the change of variables formula, it follows that a diffeomorphism ŉ is volume preserving if and only if for every measurable set A = M, µ(A) = µ(ŋ(4)). Here ...
[116]
[math/0504123] From Loop Groups to 2-Groups - arXiv
Apr 7, 2005 · We describe an interesting relation between Lie 2-algebras, the Kac-Moody central extensions of loop groups, and the group \mathrm{String}(n).
[117]
Infinite dimensional Lie algebras and current algebra - SpringerLink
The “current algebras” of elementary particle physics and quantum field theory are interpreted as infinite dimensional Lie algebras of a certain definite kind.
[118]
Lie Group-Induced Dynamics in Score-Based Generative Modeling
Sep 27, 2024 · We extend score-based generative modeling by incorporating Lie group actions on the data manifold into the denoising diffusion process.Missing: infinite- dimensional 2020s