This is a list of
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
topics, by Wikipedia page.
Examples
''See
Table of Lie groups
This article gives a table of some common Lie groups and their associated Lie algebras.
The following are noted: the topological properties of the group (dimension; connectedness; compactness; the nature of the fundamental group; and whether ...
for a list''
*
General linear group,
special linear group
**
SL2(R)
**
SL2(C)
*
Unitary group,
special unitary group
**
SU(2)
**
SU(3)
*
Orthogonal group
In mathematics, the orthogonal group in dimension , denoted , is the Group (mathematics), group of isometry, distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by ...
,
special orthogonal group
In mathematics, the orthogonal group in dimension , denoted , is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. T ...
**
Rotation group SO(3)
**
SO(8)
**
Generalized orthogonal group
In mathematics, the indefinite orthogonal group, is the Lie group of all linear transformations of an ''n''- dimensional real vector space that leave invariant a nondegenerate, symmetric bilinear form of signature , where . It is also called the ...
,
generalized special orthogonal group
In mathematics, the indefinite orthogonal group, is the Lie group of all linear transformations of an ''n''-dimension of a vector space, dimensional real vector space that leave invariant a nondegenerate form, nondegenerate, symmetric bilinear for ...
***The special unitary group SU(1,1) is the unit sphere in the ring of
coquaternion
In abstract algebra, the split-quaternions or coquaternions form an algebraic structure introduced by James Cockle in 1849 under the latter name. They form an associative algebra of dimension four over the real numbers.
After introduction in ...
s. It is the group of
hyperbolic motions of the Poincaré disk model of the
Hyperbolic plane.
***
Lorentz group
**
Spinor group
In mathematics the spin group Spin(''n'') page 15 is the double cover of the special orthogonal group , such that there exists a short exact sequence of Lie groups (when )
:1 \to \mathrm_2 \to \operatorname(n) \to \operatorname(n) \to 1.
As a Li ...
*
Symplectic group
*Exceptional groups
**
G2
**
F4
**
E6
**
E7
**
E8
*
Affine group
*
Euclidean group
In mathematics, a Euclidean group is the group of (Euclidean) isometries of a Euclidean space \mathbb^n; that is, the transformations of that space that preserve the Euclidean distance between any two points (also called Euclidean transformations). ...
*
Poincaré group
*
Heisenberg group
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
s
*
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
The commutator of two elements, a ...
*
Jacobi identity
*
Universal enveloping algebra
*
Baker-Campbell-Hausdorff formula
*
Casimir invariant
*
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. Cartan's criteria (criterion of solvability and criterion of semisimplicity) show ...
*
Kac–Moody 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. Given an affine Lie algebra, one can also form the associated affine Kac-Moody al ...
*
Loop algebra
*
Graded Lie algebra
Foundational results
*
One-parameter group
In mathematics, a one-parameter group or one-parameter subgroup usually means a continuous group homomorphism
:\varphi : \mathbb \rightarrow G
from the real line \mathbb (as an additive group) to some other topological group G.
If \varphi is in ...
,
One-parameter subgroup
In mathematics, a one-parameter group or one-parameter subgroup usually means a continuous group homomorphism
:\varphi : \mathbb \rightarrow G
from the real line \mathbb (as an additive group) to some other topological group G.
If \varphi is ...
*
Matrix exponential
*
Infinitesimal transformation
*
Lie's third theorem
*
Maurer–Cartan form In mathematics, the Maurer–Cartan form for a Lie group is a distinguished differential one-form on that carries the basic infinitesimal information about the structure of . It was much used by Élie Cartan as a basic ingredient of his met ...
*
Cartan's theorem
*
Cartan's criterion
*
Local Lie group In mathematics, a pseudogroup is a set of diffeomorphisms between open sets of a space, satisfying group-like and sheaf-like properties. It is a generalisation of the concept of a group, originating however from the geometric approach of Sophus Lie ...
*
Formal group law
*
Hilbert's fifth problem
*
Hilbert-Smith conjecture
*
Lie group decompositions
*
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 ''g''0 is called a real form of a complex Lie algebra ''g'' if ''g'' is the complexification of ''g''0:
: \math ...
*
Complex Lie group
*
Complexification (Lie group)
Semisimple theory
*
Simple Lie group
*
Compact Lie group,
Compact real form
In mathematics, the notion of a real form relates objects defined over the Field (algebra), field of Real number, real and Complex number, complex numbers. A real Lie algebra ''g''0 is called a real form of a complex Lie algebra ''g'' if ''g'' is ...
*
Semisimple Lie algebra
*
Root system
*
Simply laced group
**
ADE classification
*
Maximal torus
*
Weyl group
*
Dynkin diagram
*
Weyl character formula
Representation theory
{{see also, List of representation theory topics
*
Representation of a Lie group
*
Representation of a Lie algebra
*
Adjoint representation of a Lie group
*
Adjoint representation of a Lie algebra
*
Unitary representation
*
Weight (representation theory)
*
Peter–Weyl theorem
*
Borel–Weil theorem
*
Kirillov character formula
*
Representation theory of SU(2)
*
Representation theory of SL2(R)
Applications
Physical theories
*
Pauli matrices
*
Gell-Mann matrices
*
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. Th ...
*
Noether's theorem
*
Wigner's classification
*
Gauge theory
In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie groups) ...
*
Grand unification theory
*
Supergroup Supergroup or super group may refer to:
* Supergroup (music), a music group formed by artists who are already notable or respected in their fields
* Supergroup (physics), a generalization of groups, used in the study of supersymmetry
* Supergroup ...
*
Lie superalgebra
*
Twistor theory
*
Anyon
In physics, an anyon is a type of quasiparticle that occurs only in two-dimensional systems, with properties much less restricted than the two kinds of standard elementary particles, fermions and bosons. In general, the operation of exchangi ...
*
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 ...
*
Virasoro algebra
Geometry
*
Erlangen programme
*
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 transitively. The elements of ' ...
**
Principal homogeneous space
*
Invariant theory
*
Lie derivative
*
Darboux derivative The Darboux derivative of a map between a manifold and a Lie group is a variant of the standard derivative. It is arguably a more natural generalization of the single-variable derivative. It allows a generalization of the single-variable fundament ...
*
Lie groupoid
*
Lie algebroid
Discrete groups
*
Lattice (group)
*
Lattice (discrete subgroup)
*
Frieze group
*
Wallpaper group
*
Space group
*
Crystallographic group
*
Fuchsian group
*
Modular group
In mathematics, the modular group is the projective special linear group of matrices with integer coefficients and determinant 1. The matrices and are identified. The modular group acts on the upper-half of the complex plane by fractional l ...
*
Congruence subgroup
*
Kleinian group
*
Discrete Heisenberg group
In mathematics, the Heisenberg group H, named after Werner Heisenberg, is the group of 3×3 upper triangular matrices of the form
::\begin
1 & a & c\\
0 & 1 & b\\
0 & 0 & 1\\
\end
under the operation of matrix multiplication. Element ...
*
Clifford–Klein form In mathematics, a Clifford–Klein form is a double coset space
:Γ\''G''/''H'',
where ''G'' is a reductive Lie group, ''H'' a closed subgroup of ''G'', and Γ a discrete subgroup of G that acts properly discontinuously on the homogeneous space ' ...
Algebraic groups
*
Borel subgroup
*
Parabolic subgroup
*
Arithmetic group
Special functions
*
Dunkl operator
In mathematics, particularly the study of Lie groups, a Dunkl operator is a certain kind of mathematical operator, involving differential operators but also reflections in an underlying space.
Formally, let ''G'' be a Coxeter group with reduced ro ...
Automorphic forms
*
Modular form
In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the Group action (mathematics), group action of the modular group, and also satisfying a grow ...
*
Langlands program
People
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Sophus Lie (1842 – 1899)
*
Wilhelm Killing (1847 – 1923)
*
Élie Cartan (1869 – 1951)
*
Hermann Weyl
Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he is assoc ...
(1885 – 1955)
*
Harish-Chandra (1923 – 1983)
*
Lajos Pukánszky Lajos Pukánszky (1928-1996) 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 ...
(1928 – 1996)
*
Bertram Kostant (1928 – 2017)
Lie groups
Lie algebras
Lie groups
Lie groups