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Algebraic structure used in analysis
mathematics, a Lie algebra (pronounced /liː/ LEE) is a vector space g {\displaystyle {\mathfrak {g}}} together with an operation called the Lie bracket, an
Lie_algebra
Connected non-abelian Lie group lacking nontrivial connected normal subgroups
used to read off the list of simple Lie algebras and Riemannian symmetric spaces. Together with the commutative Lie group of the real numbers, R {\displaystyle
Simple_Lie_group
Type of Kac–Moody algebras
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
Affine_Lie_algebra
Direct sum of simple Lie algebras
mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero
Semisimple_Lie_algebra
Group that is also a differentiable manifold with group operations that are smooth
matrix Lie algebra, there is a linear group (matrix Lie group) with this algebra as its Lie algebra. On the other hand, Lie groups with isomorphic Lie algebras
Lie_group
Concept in Lie algebra mathematics
In algebra, a simple Lie algebra is a Lie algebra that is non-abelian and contains no nonzero proper ideals. The classification of real simple Lie algebras
Simple_Lie_algebra
In mathematics, a type of algebra
Lie algebra g {\displaystyle {\mathfrak {g}}} is solvable if its derived series terminates in the zero subalgebra. The derived Lie algebra of the Lie
Solvable_Lie_algebra
Representation of a Lie algebra as a set of linear transformations
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 (or endomorphisms
Lie_algebra_representation
Correspondence between topics in Lie theory
In mathematics, Lie group–Lie algebra correspondence allows one to correspond a Lie group to a Lie algebra or vice versa, and study the conditions for
Lie group–Lie algebra correspondence
Lie_group–Lie_algebra_correspondence
In mathematics, a free Lie algebra over a field K is a Lie algebra generated by a set X, without any imposed relations other than the defining relations
Free_Lie_algebra
Sum of elements on the main diagonal
{\displaystyle K} ) to the Lie algebra K of scalars; as K is Abelian (the Lie bracket vanishes), the fact that this is a map of Lie algebras is exactly the statement
Trace_(linear_algebra)
248-dimensional exceptional simple Lie group
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
E8_(mathematics)
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
Graded_Lie_algebra
Branch of mathematics
In mathematics, a Lie algebra g {\displaystyle {\mathfrak {g}}} is nilpotent if its lower central series terminates in the zero subalgebra. The lower
Nilpotent_Lie_algebra
In the mathematical field of Lie theory, the radical of a Lie algebra g {\displaystyle {\mathfrak {g}}} is the largest solvable ideal of g . {\displaystyle
Radical_of_a_Lie_algebra
Cohomology theory for Lie algebras
mathematics, Lie algebra cohomology is a cohomology theory for Lie algebras. It was first introduced in 1929 by Élie Cartan to study the topology of Lie groups
Lie_algebra_cohomology
Concept in mathematics
enveloping algebra of a Lie algebra is the unital associative algebra whose representations correspond precisely to the representations of that Lie algebra. Universal
Universal_enveloping_algebra
Group in group theory and physics
constants forms a Lie algebra under the Poisson bracket. This Lie algebra is a one-dimensional central extension of the commutative Lie algebra R 2 n {\displaystyle
Heisenberg_group
Algebraic structure
In mathematics, a Lie algebra has been generalized in several ways. A graded Lie algebra is a Lie algebra with grading. When the grading is Z / 2 {\displaystyle
Generalization of a Lie algebra
Generalization_of_a_Lie_algebra
The classical Lie algebras are finite-dimensional Lie algebras over a field which can be classified into four types A n {\displaystyle A_{n}} , B n {\displaystyle
Classical_Lie_algebras
Branch of mathematics that studies abstract algebraic structures
matrix multiplication). The algebraic objects amenable to such a description include groups, associative algebras and Lie algebras. The most prominent of these
Representation_theory
Mathematical theory of topological spaces
the two algebraic descriptions of the rational homotopy category. In short, a Lie algebra determines a graded-commutative algebra by Lie algebra cohomology
Rational_homotopy_theory
Ring that is also a vector space or a module
commutative algebra. The universal enveloping algebra of a Lie algebra is an associative algebra that can be used to study the given Lie algebra. If G is
Associative_algebra
Algebraic construct of interest in theoretical physics
class of Hopf algebra. The same term is also used for other Hopf algebras that deform or are close to classical Lie groups or Lie algebras, such as a "bicrossproduct"
Quantum_group
Nilpotent subalgebra of a Lie algebra
CSA, is a nilpotent subalgebra h {\displaystyle {\mathfrak {h}}} of a Lie algebra g {\displaystyle {\mathfrak {g}}} that is self-normalising (if [ X ,
Cartan_subalgebra
Algebraic structure used in theoretical physics
mathematics, a Lie superalgebra is a generalisation of a Lie algebra to include a Z / 2 Z {\displaystyle \mathbb {Z} /2\mathbb {Z} } ‑grading. Lie superalgebras
Lie_superalgebra
Concept in mathematics
In mathematics, the special linear Lie algebra of order n {\displaystyle n} over a field F {\displaystyle F} , denoted s l n F {\displaystyle {\mathfrak
Special_linear_Lie_algebra
Creating a "larger" Lie algebra from a smaller one, in one of several ways
Lie groups, Lie algebras and their representation theory, a Lie algebra extension e is an enlargement of a given Lie algebra g by another Lie algebra
Lie_algebra_extension
Group representation
being the use of the corresponding 'infinitesimal' representations of Lie algebras. A complex representation of a group is an action by a group on a finite-dimensional
Representation_of_a_Lie_group
Complex simple Lie Algebra
In mathematics, an exceptional Lie algebra is a complex simple Lie algebra whose Dynkin diagram is of exceptional (nonclassical) type. There are exactly
Exceptional_Lie_algebra
Lie algebra, usually infinite-dimensional
a Kac–Moody algebra (named for Victor Kac and Robert Moody, who independently and simultaneously discovered them in 1968) is a Lie algebra, usually infinite-dimensional
Kac–Moody_algebra
In mathematics, a Malcev Lie algebra, or Mal'tsev Lie algebra, is a generalization of a rational nilpotent Lie algebra, and Malcev groups are similar
Malcev_Lie_algebra
Algebra over a field where binary multiplication is not necessarily associative
unital, but Lie algebras never are. The nonassociative algebra structure of A may be studied by associating it with other associative algebras which are
Non-associative_algebra
Mathematical term
adjoint action) of a Lie group G is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a
Adjoint_representation
Generalization of a Lie algebra
A Lie conformal algebra is in some sense a generalization of a Lie algebra in that it too is a "Lie algebra," though in a different pseudo-tensor category
Lie_conformal_algebra
Homomorphisms between simple modules over the same ring are isomorphisms or zero
theory of finite groups. Schur's lemma admits generalisations to Lie groups and Lie algebras, the most common of which are due to Jacques Dixmier and Daniel
Schur's_lemma
Study of Lie groups, Lie algebras and differential equations
The foundation of Lie theory is the exponential map relating Lie algebras to Lie groups which is called the Lie group–Lie algebra correspondence. The
Lie_theory
mathematical theories of Lie groups and Lie algebras. For the topics in the representation theory of Lie groups and Lie algebras, see Glossary of representation
Glossary of Lie groups and Lie algebras
Glossary_of_Lie_groups_and_Lie_algebras
representation theory of semisimple Lie algebras is one of the crowning achievements of the theory of Lie groups and Lie algebras. The theory was worked out mainly
Representation theory of semisimple Lie algebras
Representation_theory_of_semisimple_Lie_algebras
In mathematics, a restricted Lie algebra (or p-Lie algebra) is a Lie algebra over a field of characteristic p>0 together with an additional "pth power"
Restricted_Lie_algebra
Type of derivative in differential geometry
Lie algebra with respect to this Lie bracket. The Lie derivative constitutes an infinite-dimensional Lie algebra representation of this Lie algebra,
Lie_derivative
Mathematical description of fermions
The basis elements of so(3,1) are labeled Mμν. A representation of the Lie algebra so(3,1) of the Lorentz group O(3,1) will emerge among matrices that will
Dirac_spinor
Geometric arrangements of points, foundational to Lie theory
the theory of Lie groups and Lie algebras, especially the classification and representation theory of semisimple Lie algebras. Since Lie groups (and some
Root_system
quasi-Frobenius Lie algebra ( g , [ , ] , β ) {\displaystyle ({\mathfrak {g}},[\,\,\,,\,\,\,],\beta )} over a field k {\displaystyle k} is a Lie algebra ( g , [
Quasi-Frobenius_Lie_algebra
Mathematical theory
field of Lie theory, there are two definitions of a compact Lie algebra. Extrinsically and topologically, a compact Lie algebra is the Lie algebra of a compact
Compact_Lie_algebra
Group of unitary complex matrices with determinant of 1
Lie algebra s u ( n ) {\displaystyle {\mathfrak {su}}(n)} of SU(n) consists of n × n skew-Hermitian matrices with trace zero. This (real) Lie algebra
Special_unitary_group
Universal construction of a complex Lie group from a real Lie group
is unique up to unique isomorphism. Its Lie algebra is a quotient of the complexification of the Lie algebra of the original group. They are isomorphic
Complexification_(Lie_group)
mathematics, in particular abstract algebra and topology, a homotopy Lie algebra (or L ∞ {\displaystyle L_{\infty }} -algebra) is a generalisation of the concept
Homotopy_Lie_algebra
Mathematical operation on vectors in 3D space
product is an algebra over the real numbers, which is neither commutative nor associative, but is a Lie algebra with the cross product being the Lie bracket
Cross_product
In the mathematical field of Lie theory, a split Lie algebra is a pair ( g , h ) {\displaystyle ({\mathfrak {g}},{\mathfrak {h}})} where g {\displaystyle
Split_Lie_algebra
Linear function satisfying a support condition
be an algebraically closed field and G a linear algebraic group (that is, affine algebraic group) over k. By definition, Lie(G) is the Lie algebra of all
Distribution on a linear algebraic group
Distribution_on_a_linear_algebraic_group
Concept in mathematics
2015 Proposition 4.8 Hall 2015 Section 4.4 Hall, Brian C. (2015), Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Graduate Texts
Unitary_representation
Mathematical group
\mathbb {F} )} is considered a simple Lie group. The real rank of the corresponding Lie algebra, and hence of the Lie group Sp ( 2 n , F ) {\displaystyle
Symplectic_group
In mathematics, a Lie-* algebra is a D-module with a Lie* bracket. They were introduced by Alexander Beilinson and Vladimir Drinfeld, and are similar to
Lie-*_algebra
Manifold with inversion symmetry
{k}}\oplus {\mathfrak {m}},}} where k {\displaystyle {\mathfrak {k}}} , the Lie algebra of K, is the +1 eigenspace of σ and m {\displaystyle {\mathfrak {m}}}
Hermitian_symmetric_space
Matrices named after Élie Cartan
mathematician Élie Cartan. Amusingly, the Cartan matrices in the context of Lie algebras were first investigated by Wilhelm Killing, whereas the Killing form
Cartan_matrix
In mathematics, a Malcev algebra (or Maltsev algebra or Moufang–Lie algebra) over a field is a nonassociative algebra that is antisymmetric, so that x
Malcev_algebra
133-dimensional exceptional simple Lie group
mathematics, E7 is the name of several closely related Lie groups, linear algebraic groups or their Lie algebras e7, all of which have dimension 133; the same
E7_(mathematics)
Infinite-dimensional generalized Kac-Moody algebra
In mathematics, the monster Lie algebra is an infinite-dimensional generalized Kac–Moody algebra acted on by the monster group, which was used to prove
Monster_Lie_algebra
systems and Dynkin diagrams, strikingly similar to those of semisimple Lie algebras. A comprehensive introduction is found in the lecture of Heckenberger
Nichols_algebra
Type of group in mathematics
whose inverse equals its transpose). The orthogonal group is an algebraic group and a Lie group. It is compact. The orthogonal group in dimension n has
Orthogonal_group
In mathematics, a complex Lie algebra is a Lie algebra over the complex numbers. Given a complex Lie algebra g {\displaystyle {\mathfrak {g}}} , its conjugate
Complex_Lie_algebra
finite-dimensional, semisimple Lie algebras in characteristic zero. Historically, they are regarded as leading to the discovery of Lie algebra cohomology. One usually
Whitehead's lemma (Lie algebra)
Whitehead's_lemma_(Lie_algebra)
theorem Lie algebra Lie-* algebra Lie algebra bundle Lie algebra cohomology Lie algebra representation Lie algebroid Lie bialgebra Lie coalgebra Lie conformal
List of things named after Sophus Lie
List_of_things_named_after_Sophus_Lie
Concept in Lie algebra representation theory
from its application to representations of Lie algebras and hence also to representations of algebraic and Lie groups. In this context, a weight of a representation
Weight (representation theory)
Weight_(representation_theory)
Formula in Lie theory
the Lie algebra of a Lie group. There are various ways of writing the formula, but all ultimately yield an expression for Z {\displaystyle Z} in Lie algebraic
Baker–Campbell–Hausdorff formula
Baker–Campbell–Hausdorff_formula
In mathematics, a pre-Lie algebra is an algebraic structure on a vector space that describes some properties of objects such as rooted trees and vector
Pre-Lie_algebra
Map from a Lie algebra to its Lie group
In the theory of Lie groups, the exponential map is a map from the Lie algebra g {\displaystyle {\mathfrak {g}}} of a Lie group G {\displaystyle G} to
Exponential_map_(Lie_theory)
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: g ≃ g
Real_form_(Lie_theory)
mathematics, a modular Lie algebra is a Lie algebra over a field of positive characteristic. The theory of modular Lie algebras is significantly different
Modular_Lie_algebra
Lie group of Lorentz transformations
matrix Lie group, its corresponding Lie algebra s o ( 1 , 3 ) {\displaystyle {\mathfrak {so}}(1,3)} is a matrix Lie algebra, which may be computed as s o (
Lorentz_group
In abstract algebra, specifically the theory of Lie algebras, Serre's theorem states: given a (finite reduced) root system Φ {\displaystyle \Phi } , there
Serre's theorem on a semisimple Lie algebra
Serre's_theorem_on_a_semisimple_Lie_algebra
Concept in topology (mathematics)
In mathematics, a weak Lie algebra bundle ξ = ( ξ , p , X , θ ) {\displaystyle \xi =(\xi ,p,X,\theta )\,} is a vector bundle ξ {\displaystyle \xi \,}
Lie_algebra_bundle
Pictorial representation of symmetry
Dynkin diagrams arise in the classification of semisimple Lie algebras over algebraically closed fields, in the classification of Weyl groups and other
Dynkin_diagram
In algebra, let g be a Lie algebra over a field K. Let further ξ ∈ g ∗ {\displaystyle \xi \in {\mathfrak {g}}^{*}} be a one-form on g. The stabilizer
Index_of_a_Lie_algebra
mathematics, a Lie algebra is reductive if its adjoint representation is completely reducible, hence the name. More concretely, a Lie algebra is reductive
Reductive_Lie_algebra
Distinguished element of a Lie algebra's center
distinguished element of the center of the universal enveloping algebra of a Lie algebra. A prototypical example is the squared angular momentum operator
Casimir_element
low-dimensional real Lie algebras, published in Russian in 1963. It complements the article on Lie algebra in the area of abstract algebra. An English version
Classification of low-dimensional real Lie algebras
Classification_of_low-dimensional_real_Lie_algebras
Type of 2D conformal field theory
associated to a Lie group (or supergroup), and its symmetry algebra is the affine Lie algebra built from the corresponding Lie algebra (or Lie superalgebra)
Wess–Zumino–Witten_model
Lie groups and their associated Lie algebras
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
Table_of_Lie_groups
In algebra, a linear Lie algebra is a subalgebra g {\displaystyle {\mathfrak {g}}} of the Lie algebra g l ( V ) {\displaystyle {\mathfrak {gl}}(V)} consisting
Linear_Lie_algebra
Mathematical function, denoted exp(x) or e^x
to accept other types of arguments, such as matrices and elements of Lie algebras. The graph of y = e x {\displaystyle y=e^{x}} is upward-sloping, and
Exponential_function
Representation theory
representation of a semisimple Lie algebra. In Weyl's approach to the representation theory of connected compact Lie groups, the proof of the character
Weyl_character_formula
Special kind of square matrix
{\displaystyle {\mathfrak {n}}.} This algebra is the derived Lie algebra of b {\displaystyle {\mathfrak {b}}} , the Lie algebra of all upper triangular matrices;
Triangular_matrix
Proof that every structure with certain properties is isomorphic to another structure
enveloping algebra. Ado's theorem states that every finite-dimensional Lie algebra over a field of characteristic zero embeds into the Lie algebra of endomorphisms
Representation_theorem
In mathematics, the Hecke algebra of a pair (G, K) of locally compact or reductive Lie groups is an algebra of measures under convolution. It can also
Hecke_algebra_of_a_pair
In mathematics, especially in Lie theory, En is the Kac–Moody algebra whose Dynkin diagram is a bifurcating graph with three branches of length 1, 2 and
En_(Lie_algebra)
Algebra used in 2D conformal field theories and string theory
notion of vertex algebra was introduced by Richard Borcherds in 1986, motivated by a construction of an infinite-dimensional Lie algebra due to Igor Frenkel
Vertex_operator_algebra
Representation of the symmetry group of spacetime in special relativity
irreducible representations of the Lie algebra of the Lorentz group can be derived by factoring that Lie algebra into a direct product of two subalgebras
Representation theory of the Lorentz group
Representation_theory_of_the_Lorentz_group
In algebra, the nilradical of a Lie algebra is a nilpotent ideal, which is as large as possible. The nilradical n i l ( g ) {\displaystyle {\mathfrak {nil}}({\mathfrak
Nilradical_of_a_Lie_algebra
Subgroup of a root system's isometry group
In mathematics, in particular the theory of Lie algebras, the Weyl group (named after Hermann Weyl) of a root system Φ is a subgroup of the isometry group
Weyl_group
Real square matrix whose columns and rows are orthogonal unit vectors
In linear algebra, an orthogonal matrix or orthonormal matrix Q, is a real-valued square matrix whose columns and rows are orthonormal vectors. One way
Orthogonal_matrix
Mathematical concept
appear prominently in the classification of Lie groups. Analogous terms are used for Lie algebras (using the Lie bracket) including nilpotent, lower central
Nilpotent_group
to finding a (quantum) algebra whose classical limit is a given (classical) algebra such as a Lie algebra or a Poisson algebra. Intuitively, a deformation
Deformation_quantization
Associative algebra together with a Lie bracket that satisfies Leibniz's law
In mathematics, a Poisson algebra is an associative algebra together with a Lie bracket that also satisfies Leibniz's law; that is, the bracket is also
Poisson_algebra
Operation in Hamiltonian mechanics
as well: it occurs in the theory of Lie algebras, where the tensor algebra of a Lie algebra forms a Poisson algebra; a detailed construction of how this
Poisson_bracket
Such an algebra is called a Lie superalgebra. Just as one can have representations of a Lie algebra, one can also have representations of a Lie superalgebra
Supersymmetry_algebra
Construction in algebra
Quasitriangular Hopf algebra Algebra/set analogy Representation theory of Hopf algebras Ribbon Hopf algebra Superalgebra Supergroup Anyonic Lie algebra Sweedler's
Hopf_algebra
Family of linear transformations
generators being a basis of the Lie algebra in the usual vector space sense. The exponential map from the Lie algebra to the Lie group, exp : s o ( 3 , 1 )
Lorentz_transformation
In algebra, Weyl's theorem on complete reducibility is a fundamental result in the theory of Lie algebra representations (specifically in the representation
Weyl's theorem on complete reducibility
Weyl's_theorem_on_complete_reducibility
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