Examples

• General linear group
  General linear group
  In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible...
  
  , special linear group
  Special linear group
  In mathematics, the special linear group of degree n over a field F is the set of n×n matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion....
  
  
  • SL₂(R)
  • SL₂(C)
• Unitary group
  Unitary group
  In mathematics, the unitary group of degree n, denoted U, is the group of n×n unitary matrices, with the group operation that of matrix multiplication. The unitary group is a subgroup of the general linear group GL...
  
  , special unitary group
  Special unitary group
  The special unitary group of degree n, denoted SU, is the group of n×n unitary matrices with determinant 1. The group operation is that of matrix multiplication...
  
  
  • SU(2)
  • SU(3)
• Orthogonal group
  Orthogonal group
  In mathematics, the orthogonal group of degree n over a field F is the group of n × n orthogonal matrices with entries from F, with the group operation of matrix multiplication...
  
  , special orthogonal group
  • Rotation group
    Rotation group
    In mechanics and geometry, the rotation group is the group of all rotations about the origin of three-dimensional Euclidean space R3 under the operation of composition. By definition, a rotation about the origin is a linear transformation that preserves length of vectors and preserves orientation ...
    
     = SO(3)
  • SO(8)
    SO(8)
    In mathematics, SO is the special orthogonal group acting on eight-dimensional Euclidean space. It could be either a real or complex simple Lie group of rank 4 and dimension 28.-Spin:...
    
    
  • Generalized orthogonal group
    Generalized orthogonal group
    In mathematics, the indefinite orthogonal group, O is the Lie group of all linear transformations of a n = p + q dimensional real vector space which leave invariant a nondegenerate, symmetric bilinear form of signature...
    
    , generalized special orthogonal group
    • The special unitary group SU(1,1) is the unit sphere in the ring of coquaternion
      Coquaternion
      In abstract algebra, the split-quaternions or coquaternions are elements of a 4-dimensional associative algebra introduced by James Cockle in 1849 under the latter name. Like the quaternions introduced by Hamilton in 1843, they form a four dimensional real vector space equipped with a...
      
      s. It is the group of hyperbolic motion
      Hyperbolic motion
      In geometry, hyperbolic motions are isometric automorphisms of a hyperbolic space. Under composition of mappings, the hyperbolic motions form a continuous group. This group is said to characterize the hyperbolic space. Such an approach to geometry was cultivated by Felix Klein in his Erlangen program...
      
      s of the Poincaré disk model of the Hyperbolic plane
      Hyperbolic geometry
      In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced...
      
      .
    • Lorentz group
      Lorentz group
      In physics , the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical setting for all physical phenomena...
      
      
  • Spinor group
• Symplectic group
  Symplectic group
  In mathematics, the name symplectic group can refer to two different, but closely related, types of mathematical groups, denoted Sp and Sp. The latter is sometimes called the compact symplectic group to distinguish it from the former. Many authors prefer slightly different notations, usually...
  
  
• Exceptional groups
  • G₂
  • F₄
  • E₆
  • E₇
  • E₈
    E8 (mathematics)
    In mathematics, E8 is any of several closely related exceptional simple Lie groups, linear algebraic groups or Lie algebras of dimension 248; the same notation is used for the corresponding root lattice, which has rank 8...
    
    
• Affine group
  Affine group
  In mathematics, the affine group or general affine group of any affine space over a field K is the group of all invertible affine transformations from the space into itself.It is a Lie group if K is the real or complex field or quaternions....
  
  
• Euclidean group
  Euclidean group
  In mathematics, the Euclidean group E, sometimes called ISO or similar, is the symmetry group of n-dimensional Euclidean space...
  
  
• Poincaré group
  Poincaré group
  In physics and mathematics, the Poincaré group, named after Henri Poincaré, is the group of isometries of Minkowski spacetime.-Simple explanation:...
  
  
• Heisenberg group

Lie algebra

Lie algebra

In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...

s

• Commutator
  Commutator
  In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.-Group theory:...
  
  
• Jacobi identity
  Jacobi identity
  In mathematics the Jacobi identity is a property that a binary operation can satisfy which determines how the order of evaluation behaves for the given operation. Unlike for associative operations, order of evaluation is significant for operations satisfying Jacobi identity...
  
  
• Universal enveloping algebra
  Universal enveloping algebra
  In mathematics, for any Lie algebra L one can construct its universal enveloping algebra U. This construction passes from the non-associative structure L to a unital associative algebra which captures the important properties of L.Any associative algebra A over the field K becomes a Lie algebra...
  
  
• Campbell-Hausdorff formula
• Casimir invariant
  Casimir invariant
  In mathematics, a Casimir invariant or Casimir operator is a distinguished element of the centre of the universal enveloping algebra of a Lie algebra...
  
  
• Killing form
  Killing form
  In mathematics, the Killing form, named after Wilhelm Killing, is a symmetric bilinear form that plays a basic role in the theories of Lie groups and Lie algebras...
  
  
• Kac-Moody algebra
• Affine Lie algebra
  Affine Lie algebra
  In mathematics, an affine Lie algebra is an infinite-dimensional Lie algebra that is constructed in a canonical fashion out of a finite-dimensional simple Lie algebra. It is a Kac–Moody algebra for which the generalized Cartan matrix is positive semi-definite and has corank 1...
  
  
• Loop algebra
• Graded Lie algebra
  Graded Lie algebra
  In mathematics, a graded Lie algebra is a Lie algebra endowed with a gradation which is compatible with the Lie bracket. In other words, a graded Lie algebra is a Lie algebra which is also a nonassociative graded algebra under the bracket operation. A choice of Cartan decomposition endows any...
  
  

Foundational results

• One-parameter group
  One-parameter group
  In mathematics, a one-parameter group or one-parameter subgroup usually means a continuous group homomorphismfrom the real line R to some other topological group G...
  
  , One-parameter subgroup
• Matrix exponential
  Matrix exponential
  In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. Abstractly, the matrix exponential gives the connection between a matrix Lie algebra and the corresponding Lie group....
  
  
• Infinitesimal transformation
  Infinitesimal transformation
  In mathematics, an infinitesimal transformation is a limiting form of small transformation. For example one may talk about an infinitesimal rotation of a rigid body, in three-dimensional space. This is conventionally represented by a 3×3 skew-symmetric matrix A...
  
  
• Lie's third theorem
  Lie's third theorem
  In mathematics, Lie's third theorem often means the result that states that any finite-dimensional Lie algebra g, over the real numbers, is the Lie algebra associated to some Lie group G. The relationship to the history has though become confused....
  
  
• Maurer-Cartan form
  Maurer-Cartan form
  In mathematics, the Maurer–Cartan form for a Lie group G is a distinguished differential one-form on G that carries the basic infinitesimal information about the structure of G...
  
  
• Cartan's theorem
  Cartan's theorem
  In mathematics, three results in Lie group theory are called Cartan's theorem, named after Élie Cartan:See also Cartan's theorems A and B, results of Henri Cartan, and Cartan's lemma for various other results attributed to Élie and Henri Cartan....
  
  
• Cartan's criterion
• Local Lie group
• Formal group law
• Hilbert's fifth problem
  Hilbert's fifth problem
  Hilbert's fifth problem, is the fifth mathematical problem from the problem-list publicized in 1900 by mathematician David Hilbert, and concerns the characterization of Lie groups. The theory of Lie groups describes continuous symmetry in mathematics; its importance there and in theoretical physics...
  
  
• Hilbert-Smith conjecture
  Hilbert-Smith conjecture
  In mathematics, the Hilbert–Smith conjecture is concerned with the transformation groups of manifolds; and in particular with the limitations on topological groups G that can act effectively on a manifold M...
  
  
• Lie group decompositions
  Lie group decompositions
  In mathematics, Lie group decompositions, named after Sophus Lie, are used to analyse the structure of Lie groups and associated objects, by showing how they are built up out of subgroups. They are essential technical tools in the representation theory of Lie groups and Lie algebras; they can also...
  
  
• Real form (Lie theory)
  Real form (Lie theory)
  In mathematics, the notion of a real form relates objects defined over the field of real and complex numbers. A real Lie algebra g0 is called a real form of a complex Lie algebra g if g is the complexification of g0:...
  
  

Semisimple theory

• Simple Lie group
  Simple Lie group
  In group theory, a simple Lie group is a connected non-abelian Lie group G which does not have nontrivial connected normal subgroups.A simple Lie algebra is a non-abelian Lie algebra whose only ideals are 0 and itself...
  
  
• Compact Lie group, Compact real form
• Semisimple Lie algebra
• Root system
  Root system
  In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras...
  
  
• Simply laced group
  • ADE classification
    ADE classification
    In mathematics, the ADE classification is the complete list of simply laced Dynkin diagrams or other mathematical objects satisfying analogous axioms; "simply laced" means that there are no multiple edges, which corresponds to all simple roots in the root system forming angles of \pi/2 = 90^\circ ...
    
    
• Maximal torus
  Maximal torus
  In the mathematical theory of compact Lie groups a special role is played by torus subgroups, in particular by the maximal torus subgroups.A torus in a Lie group G is a compact, connected, abelian Lie subgroup of G . A maximal torus is one which is maximal among such subgroups...
  
  
• Weyl group
  Weyl group
  In mathematics, in particular the theory of Lie algebras, the Weyl group of a root system Φ is a subgroup of the isometry group of the root system. Specifically, it is the subgroup which is generated by reflections through the hyperplanes orthogonal to the roots, and as such is a finite reflection...
  
  
• Dynkin diagram
• Weyl character formula
  Weyl character formula
  In mathematics, the Weyl character formula in representation theory describes the characters of irreducible representations of compact Lie groups in terms of their highest weights. It was proved by ....
  
  

Representation theory

• Representation of a Lie group
  Representation of a Lie group
  In mathematics and theoretical physics, the idea of a representation of a Lie group plays an important role in the study of continuous symmetry. A great deal is known about such representations, a basic tool in their study being the use of the corresponding 'infinitesimal' representations of Lie...
  
  
• Representation of a Lie algebra
  Representation of a Lie algebra
  In the mathematical field of representation theory, a Lie algebra representation or representation of a Lie algebra is a way of writing a Lie algebra as a set of matrices in such a way that the Lie bracket is given by the commutator.The notion is closely related to that of a representation of a...
  
  
• Adjoint representation of a Lie group
• Adjoint representation of a Lie algebra
• Unitary representation
  Unitary representation
  In mathematics, a unitary representation of a group G is a linear representation π of G on a complex Hilbert space V such that π is a unitary operator for every g ∈ G...
  
  
• Weight (representation theory)
  Weight (representation theory)
  In the mathematical field of representation theory, a weight of an algebra A over a field F is an algebra homomorphism from A to F – a linear functional – or equivalently, a one dimensional representation of A over F. It is the algebra analogue of a multiplicative character of a group...
  
  
• Peter–Weyl theorem
  Peter–Weyl theorem
  In mathematics, the Peter–Weyl theorem is a basic result in the theory of harmonic analysis, applying to topological groups that are compact, but are not necessarily abelian. It was initially proved by Hermann Weyl, with his student Fritz Peter, in the setting of a compact topological group G...
  
  
• Borel–Weil theorem
  Borel–Weil theorem
  In mathematics, in the field of representation theory, the Borel–Weil theorem, named after Armand Borel and André Weil, provides a concrete model for irreducible representations of compact Lie groups and complex semisimple Lie groups. These representations are realized in the spaces of global...
  
  
• Kirillov character formula
  Kirillov character formula
  In mathematics, for a Lie group G, the Kirillov orbit method gives a heuristic method in representation theory. It connects the Fourier transforms of coadjoint orbits, which lie in the dual space of the Lie algebra of G, to the infinitesimal characters of the irreducible representations...
  
  
• Representation theory of SU(2)
  Representation theory of SU(2)
  In the study of the representation theory of Lie groups, the study of representations of SU is fundamental to the study of representations of semisimple Lie groups. It is the first case of a Lie group that is both a compact group and a non-abelian group...
  
  
• Representation theory of SL2(R)
  Representation theory of SL2(R)
  In mathematics, the main results concerning irreducible unitary representations of the Lie group SL are due to Gelfand and Naimark , V...
  
  

Physical theories

• Pauli matrices
  Pauli matrices
  The Pauli matrices are a set of three 2 × 2 complex matrices which are Hermitian and unitary. Usually indicated by the Greek letter "sigma" , they are occasionally denoted with a "tau" when used in connection with isospin symmetries...
  
  
• Gell-Mann matrices
  Gell-Mann matrices
  The Gell-Mann matrices, named for Murray Gell-Mann, are one possible representation of the infinitesimal generators of the special unitary group called SU....
  
  
• Poisson bracket
  Poisson bracket
  In mathematics and classical mechanics, the Poisson bracket is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time-evolution of a Hamiltonian dynamical system...
  
  
• Noether's theorem
  Noether's theorem
  Noether's theorem states that any differentiable symmetry of the action of a physical system has a corresponding conservation law. The theorem was proved by German mathematician Emmy Noether in 1915 and published in 1918...
  
  
• Wigner's classification
  Wigner's classification
  In mathematics and theoretical physics, Wigner's classificationis a classification of the nonnegative energy irreducible unitary representations of the Poincaré group, which have sharp mass eigenvalues...
  
  
• Gauge theory
  Gauge theory
  In physics, gauge invariance is the property of a field theory in which different configurations of the underlying fundamental but unobservable fields result in identical observable quantities. A theory with such a property is called a gauge theory...
  
  
• Grand unification theory
  Grand unification theory
  The term Grand Unified Theory, often abbreviated as GUT, refers to any of several similar candidate models in particle physics in which at high-energy, the three gauge interactions of the Standard Model which define the electromagnetic, weak, and strong interactions, are merged into one single...
  
  
• Supergroup
  Supergroup (physics)
  The concept of supergroup is a generalization of that of group. In other words, every group is a supergroup but not every supergroup is a group. A supergroup is like a Lie group in that there is a well defined notion of smooth function defined on them....
  
  
• Lie superalgebra
  Lie superalgebra
  In mathematics, a Lie superalgebra is a generalisation of a Lie algebra to include a Z2-grading. Lie superalgebras are important in theoretical physics where they are used to describe the mathematics of supersymmetry...
  
  
• Twistor
• Anyon
  Anyon
  In physics, an anyon is a type of particle that occurs only in two-dimensional systems. It is a generalization of the fermion and boson concept.-From theory to reality:...
  
  
• Witt algebra
  Witt algebra
  In mathematics, the complex Witt algebra, named after Ernst Witt, is the Lie algebra of meromorphic vector fields defined on the Riemann sphere that are holomorphic except at two fixed points. It is also the complexification of the Lie algebra of polynomial vector fields on a circle, and the Lie...
  
  
• Virasoro algebra
  Virasoro algebra
  In mathematics, the Virasoro algebra is a complex Lie algebra, given as a central extension of the complex polynomial vector fields on the circle, and is widely used in conformal field theory and string theory....
  
  

Geometry

• Erlangen programme
• Homogeneous space
  Homogeneous space
  In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group G is a non-empty manifold or topological space X on which G acts continuously by symmetry in a transitive way. A special case of this is when the topological group,...
  
  
  • Principal homogeneous space
    Principal homogeneous space
    In mathematics, a principal homogeneous space, or torsor, for a group G is a homogeneous space X for G such that the stabilizer subgroup of any point is trivial...
    
    
• Invariant theory
  Invariant theory
  Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties from the point of view of their effect on functions...
  
  
• Lie derivative
  Lie derivative
  In mathematics, the Lie derivative , named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a vector field or more generally a tensor field, along the flow of another vector field...
  
  
• Darboux derivative
  Darboux derivative
  The Darboux derivative of a map between a manifold and a Lie group is a variant of the standard derivative. In a certain sense, it is arguably a more natural generalization of the single-variable derivative...
  
  
• Lie groupoid
  Lie groupoid
  In mathematics, a Lie groupoid is a groupoid where the set Ob of objects and the set Mor of morphisms are both manifolds, the source and target operationss,t : Mor \to Ob...
  
  
• Lie algebroid
  Lie algebroid
  In mathematics, Lie algebroids serve the same role in the theory of Lie groupoids that Lie algebras serve in the theory of Lie groups: reducing global problems to infinitesimal ones...
  
  

Discrete group

Discrete group

In mathematics, a discrete group is a group G equipped with the discrete topology. With this topology G becomes a topological group. A discrete subgroup of a topological group G is a subgroup H whose relative topology is the discrete one...

s

• Lattice (group)
  Lattice (group)
  In mathematics, especially in geometry and group theory, a lattice in Rn is a discrete subgroup of Rn which spans the real vector space Rn. Every lattice in Rn can be generated from a basis for the vector space by forming all linear combinations with integer coefficients...
  
  
• Lattice (discrete subgroup)
  Lattice (discrete subgroup)
  In Lie theory and related areas of mathematics, a lattice in a locally compact topological group is a discrete subgroup with the property that the quotient space has finite invariant measure...
  
  
• Frieze group
  Frieze group
  A frieze group is a mathematical concept to classify designs on two-dimensional surfaces which are repetitive in one direction, based on the symmetries in the pattern. Such patterns occur frequently in architecture and decorative art...
  
  
• Wallpaper group
  Wallpaper group
  A wallpaper group is a mathematical classification of a two-dimensional repetitive pattern, based on the symmetries in the pattern. Such patterns occur frequently in architecture and decorative art...
  
  
• Space group
  Space group
  In mathematics and geometry, a space group is a symmetry group, usually for three dimensions, that divides space into discrete repeatable domains.In three dimensions, there are 219 unique types, or counted as 230 if chiral copies are considered distinct...
  
  
• Crystallographic group
• Fuchsian group
  Fuchsian group
  In mathematics, a Fuchsian group is a discrete subgroup of PSL. The group PSL can be regarded as a group of isometries of the hyperbolic plane, or conformal transformations of the unit disc, or conformal transformations of the upper half plane, so a Fuchsian group can be regarded as a group acting...
  
  
• Modular group
  Modular group
  In mathematics, the modular group Γ is a fundamental object of study in number theory, geometry, algebra, and many other areas of advanced mathematics...
  
  
• Congruence subgroup
  Congruence subgroup
  In mathematics, a congruence subgroup of a matrix group with integer entries is a subgroup defined by congruence conditions on the entries. A very simple example would be invertible 2x2 integer matrices of determinant 1, such that the off-diagonal entries are even.An importance class of congruence...
  
  
• Kleinian group
  Kleinian group
  In mathematics, a Kleinian group is a discrete subgroup of PSL. The group PSL of 2 by 2 complex matrices of determinant 1 modulo its center has several natural representations: as conformal transformations of the Riemann sphere, and as orientation-preserving isometries of 3-dimensional hyperbolic...
  
  
• Discrete Heisenberg group
• Clifford–Klein form

People

• Sophus Lie
  Sophus Lie
  Marius Sophus Lie was a Norwegian mathematician. He largely created the theory of continuous symmetry, and applied it to the study of geometry and differential equations.- Biography :...
  
   (1842 – 1899)
• Wilhelm Killing
  Wilhelm Killing
  Wilhelm Karl Joseph Killing was a German mathematician who made important contributions to the theories of Lie algebras, Lie groups, and non-Euclidean geometry....
  
   (1847 – 1923)
• Élie Cartan
  Élie Cartan
  Élie Joseph Cartan was an influential French mathematician, who did fundamental work in the theory of Lie groups and their geometric applications...
  
   (1869 – 1951)
• Hermann Weyl
  Hermann Weyl
  Hermann Klaus Hugo Weyl was a German mathematician and theoretical physicist. Although much of his working life was spent in Zürich, Switzerland and then Princeton, he is associated with the University of Göttingen tradition of mathematics, represented by David Hilbert and Hermann Minkowski.His...
  
   (1885 – 1955)
• Harish-Chandra
  Harish-Chandra
  Harish-Chandra was an Indian mathematician, who did fundamental work in representation theory, especially Harmonic analysis on semisimple Lie groups. -Life:...
  
   (1923 – 1983)
• Lajos Pukánszky
  Lajos Pukánszky
  Lajos Pukánszky was a Hungarian and American mathematician noted for his work in representation theory of solvable Lie groups. He was born in Budapest on November 24, 1928, defended his thesis in 1955 at the University of Szeged under Béla Szőkefalvi-Nagy, but left Hungary in 1956...
  
  (1928 – 1996)
• Bertram Kostant
  Bertram Kostant
  -Early life and education:Kostant grew up in New York City, where he graduated from the celebrated Stuyvesant High School in 1945. He went on to obtain an undergraduate degree in mathematics from Purdue University in 1950. He earned his Ph.D...