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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
Connected non-abelian Lie group lacking nontrivial connected normal subgroups
classifying the real simple Lie algebras to that of finding all the real forms of each complex simple Lie algebra (i.e., real Lie algebras whose complexification
Simple_Lie_group
Algebraic structure used in analysis
the Lie bracket measures the failure of commutativity for the Lie group.) Conversely, to any finite-dimensional Lie algebra over the real or complex numbers
Lie_algebra
Concept in Lie algebra mathematics
semisimple Lie algebra. A simple Lie group is a connected Lie group whose Lie algebra is simple. A finite-dimensional simple complex Lie algebra is isomorphic
Simple_Lie_algebra
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
Real_form_(Lie_theory)
248-dimensional exceptional simple Lie group
designation E8 comes from the Cartan–Killing classification of the complex simple Lie algebras, which fall into four infinite series labeled An, Bn, Cn, Dn
E8_(mathematics)
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
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
Universal construction of a complex Lie group from a real Lie group
in its Lie algebra. In this case the complexification is a complex algebraic group and its Lie algebra is the complexification of the Lie algebra of the
Complexification_(Lie_group)
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
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
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
split real form of a complex Lie algebra, and because split semisimple Lie algebras (more generally, split reductive Lie algebras) over any field share
Split_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
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
algebra Lie superalgebra Abelian Lie algebra Affine Lie algebra Anyonic Lie algebra Compact Lie algebra Complex Lie algebra Exceptional Lie algebra Finite-dimensional
List of things named after Sophus Lie
List_of_things_named_after_Sophus_Lie
133-dimensional exceptional simple Lie group
unique complex Lie algebra of type E7, corresponding to a complex group of complex dimension 133. The complex adjoint Lie group E7 of complex dimension
E7_(mathematics)
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
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
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
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
Lie group whose manifold is complex and whose group operation is holomorphic
of a complex Lie group. A complex semisimple Lie group is a linear algebraic group. The Lie algebra of a complex Lie group is a complex Lie algebra. A finite-dimensional
Complex_Lie_group
{\displaystyle {\mathfrak {g}}} is a *-Lie algebra (a complex Lie algebra with a * operation which is compatible with the Lie bracket), π(X) is the conjugate
Complex conjugate representation
Complex_conjugate_representation
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
Mathematical group
the integers under addition. As the real form of a simple complex Lie algebra, its Lie algebra is the split real form of s p ( 2 n , C ) {\displaystyle
Symplectic_group
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)
Group that is also a differentiable manifold with group operations that are smooth
the Lie algebra of the Lie group of positive real numbers with multiplication), for complex numbers (because C {\displaystyle \mathbb {C} } is the Lie algebra
Lie_group
Algebra describing 2D conformal symmetry
mathematics, the Virasoro algebra is a complex Lie algebra and the unique nontrivial central extension of the Witt algebra. It is widely used in two-dimensional
Virasoro_algebra
simple complex Lie algebra is given by three eigenvalues α, β, γ of the Casimir operator acting on spaces A, B, C, where the symmetric square of the Lie algebra
Vogel_plane
Algebra of meromorphic vector fields on the Riemann sphere
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
Witt_algebra
Mathematical structure in abstract algebra
associative algebra over R. Involutive algebras generalize the idea of a number system equipped with conjugation, for example the complex numbers and complex conjugation
*-algebra
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
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
Simple Lie group; the automorphism group of the octonions
mathematics, G2 is three simple Lie groups (a complex form, a compact real form and a split real form), their Lie algebras g 2 , {\displaystyle {\mathfrak
G2_(mathematics)
78-dimensional exceptional simple Lie group
unique complex Lie algebra of type E6, corresponding to a complex group of complex dimension 78. The complex adjoint Lie group E6 of complex dimension
E6_(mathematics)
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
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
abstract algebra and topology, a differential graded Lie algebra (or dg Lie algebra, or dgla) is a graded vector space with added Lie algebra and chain
Differential graded Lie algebra
Differential_graded_Lie_algebra
algebras is significantly different from the theory of real and complex Lie algebras. This difference can be traced to the properties of Frobenius automorphism
Modular_Lie_algebra
of a Lie algebra or Lie group is an element whose centralizer has dimension as small as possible. For example, in a complex semisimple Lie algebra, an
Regular element of a Lie algebra
Regular_element_of_a_Lie_algebra
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
Vector space equipped with a bilinear product
mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure
Algebra_over_a_field
Ring that is also a vector space or a module
(discrete) groups. If G is an algebraic group (e.g., semisimple complex Lie group), then the coordinate ring of G is the Hopf algebra A corresponding to G. Many
Associative_algebra
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
Group representation
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
Lie group of Lorentz transformations
the Lie algebra of the Lorentz group is isomorphic to the Lie algebra s l ( 2 , C ) {\displaystyle {\mathfrak {sl}}(2,\mathbf {C} )} . As a complex Lie
Lorentz_group
Group of unitary complex matrices with determinant of 1
set of traceless Hermitian n × n complex matrices with Lie bracket given by −i times the commutator. The Lie algebra s u ( n ) {\displaystyle {\mathfrak
Special_unitary_group
Symmetric bilinear form in mathematics
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
Killing_form
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
Manifold with inversion symmetry
obtained by replacing H with the closed real Lie subgroup H* of the complex Lie group G with Lie algebra h ∗ = k ⊕ i m ⊂ g . {\displaystyle {\mathfrak
Hermitian_symmetric_space
In algebra, a parabolic Lie algebra p {\displaystyle {\mathfrak {p}}} is a subalgebra of a semisimple Lie algebra g {\displaystyle {\mathfrak {g}}} satisfying
Parabolic_Lie_algebra
In mathematics, a Chevalley basis for a simple complex Lie algebra is a basis constructed by Claude Chevalley with the property that all structure constants
Chevalley_basis
Type of Lie algebra of interest in physics
In mathematics, loop algebras are certain types of Lie algebras, of particular interest in theoretical physics. For a Lie algebra g {\displaystyle {\mathfrak
Loop_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
Type of algebras, possibly non associative
In mathematics, a composition algebra A over a field K is a not necessarily associative algebra over K together with a nondegenerate quadratic form N
Composition_algebra
Algebra based on a vector space with a quadratic form
the Lie algebra so(1, 3) sitting inside the Clifford algebra conventionally requires a complex Clifford algebra. For reference, the spin Lie algebra is
Clifford_algebra
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
52-dimensional exceptional simple Lie group
In mathematics, F4 is a Lie group and also its Lie algebra f4. It is one of the five exceptional simple Lie groups. F4 has rank 4 and dimension 52. The
F4_(mathematics)
Group of 𝑛 × 𝑛 invertible matrices
positive determinant. This is also a Lie group of dimension n 2 {\displaystyle n^{2}} ; it has the same Lie algebra as GL ( n , R ) {\displaystyle \operatorname
General_linear_group
Describes the objects of a given type, up to some equivalence
abstract algebra Classification of low-dimensional real Lie algebras Classification of Simple Lie algebras and groups Classification of simple complex Lie algebras –
Classification_theorem
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
Formula in representation theory
semisimple complex Lie algebra in a tensor product of two irreducible representations. It is a consequence of the Weyl character formula, and for the Lie algebra
Steinberg_formula
Theorem in the mathematics of Lie's theory
In the mathematics of Lie theory, Lie's third theorem states that every finite-dimensional Lie algebra g {\displaystyle {\mathfrak {g}}} over the real
Lie's_third_theorem
Sum of elements on the main diagonal
])=\operatorname {tr} ([\mathbf {X} ,\mathbf {Y} ]\mathbf {Z} ).} For a complex simple Lie algebra (such as s l {\displaystyle {\mathfrak {sl}}} n), every such bilinear
Trace_(linear_algebra)
Branch of mathematics that studies abstract algebraic structures
real or complex numbers. There are three main sorts of algebraic objects for which this can be done: groups, associative algebras and Lie algebras. The set
Representation_theory
Supersymmetric generalization of the Poincaré algebra
of supersymmetry algebras (without central charges or internal symmetries), and are Lie superalgebras. Thus a super-Poincaré algebra is a Z2-graded vector
Super-Poincaré_algebra
Concept in Lie algebra representation theory
of a derivation.) Let g {\displaystyle {\mathfrak {g}}} be a complex semisimple Lie algebra and h {\displaystyle {\mathfrak {h}}} a Cartan subalgebra of
Weight (representation theory)
Weight_(representation_theory)
Branch of mathematics
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are sets with specific operations
Abstract_algebra
representation theory of real and complex semisimple Lie groups and semisimple Lie algebras. An element X of a semisimple Lie algebra g is called nilpotent if
Nilpotent_orbit
Type of automorphism
In abstract algebra, an automorphism of a Lie algebra g {\displaystyle {\mathfrak {g}}} is an isomorphism from g {\displaystyle {\mathfrak {g}}} to itself
Automorphism_of_a_Lie_algebra
Every polynomial has a real or complex root
of algebra, also called d'Alembert's theorem or the d'Alembert–Gauss theorem, states that every non-constant single-variable polynomial with complex coefficients
Fundamental theorem of algebra
Fundamental_theorem_of_algebra
e2L(a), the corresponding complex Lie algebra contains the operators L(a). The commutators [L(a),L(b)] span the complex Lie algebra of derivations of A. The
Mutation_(Jordan_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
gives conditions for a Lie algebra in characteristic 0 to be solvable, which implies a related criterion for the Lie algebra to be semisimple. It is
Cartan's_criterion
enveloping algebra Baker–Campbell–Hausdorff formula Casimir invariant Killing form Kac–Moody algebra Affine Lie algebra Loop algebra Graded Lie algebra One-parameter
List_of_Lie_groups_topics
any Lie algebra is obviously a Leibniz algebra. In this sense, Leibniz algebras can be seen as a non-commutative generalization of Lie algebras. The
Leibniz_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
Mathematical group
group of Lie type usually refers to finite groups that are closely related to the group of rational points of a reductive linear algebraic group with
Group_of_Lie_type
mathematics a Lie coalgebra is the dual structure to a Lie algebra. In finite dimensions, these are dual objects: the dual vector space to a Lie algebra naturally
Lie_coalgebra
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
Theorem in representation theory
highest weight classifies the irreducible representations of a complex semisimple Lie algebra g {\displaystyle {\mathfrak {g}}} . There is a closely related
Theorem_of_the_highest_weight
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
French mathematician (1869–1951)
proving the existence of the exceptional Lie algebras belonging to each of the types of simple complex Lie algebras that Killing had shown to be possible
Élie_Cartan
Branch of mathematics
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants
Algebraic_topology
Lie algebra classification
complex Lie algebras. Dimension 0: The only Lie algebra is the abelian Lie algebra R0. Dimension 1: The only Lie algebra is the abelian Lie algebra R1
Bianchi_classification
Not-necessarily-associative commutative algebra satisfying (xy)(xx) = x(y(xx))
automorphism group is the exceptional Lie group F4. Since over the complex numbers this is the only simple exceptional Jordan algebra up to isomorphism, it is often
Jordan_algebra
Term in mathematics
In the mathematical study of Lie algebras and Lie groups, Satake diagrams are a generalization of Dynkin diagrams that classify involutions of root systems
Satake_diagram
Algebraic variety with a group structure
Similarly to the Lie group–Lie algebra correspondence, to an algebraic group over a field k {\displaystyle k} is associated a Lie algebra over k {\displaystyle
Algebraic_group
Study of abstract algebraic structures
In abstract algebra, a representation of an associative algebra is a module for that algebra. Here an associative algebra is a (not necessarily unital)
Algebra_representation
Physics property associated with symmetries
(Lorentz) scalar. Note that the complex Lie algebra sl(2,C) has a compact real form su(2) (in fact, all Lie algebras have a unique compact real form)
Charge_(physics)
{\displaystyle {\mathfrak {g}}} is the Lie algebra of a complex Lie group, then a Borel subalgebra is the Lie algebra of a Borel subgroup. Let g = g l ( V
Borel_subalgebra
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
Non-tensorial representation of the spin group
irreducible modules, each of dimension 2k. The Lie algebra so(V, g) embeds in the even part of the Clifford algebra, equipped with the commutator bracket, and
Spinor
Infinite dimensional Lie algebra occurring in quantum field theory
Lie algebra called a current algebra. Mathematically these are Lie algebras consisting of smooth maps from a manifold into a finite dimensional Lie algebra
Current_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
theorem (Lie algebra) Levi's theorem (Lie groups) Lie's theorem (Lie algebra) Poincaré–Birkhoff–Witt theorem (universal enveloping algebras) Shirshov–Cohn
List_of_theorems
Mapping between categories
action is a group homomorphism. Lie algebras Assigning to every real (complex) Lie group its real (complex) Lie algebra defines a functor. Tensor products
Functor
Topics referred to by the same term
algebra Symplectic geometry Symplectic group, and corresponding symplectic Lie algebra Symplectic integrator Symplectic manifold Symplectic matrix Symplectic
Symplectic
Space with one dimension
action of T. In Lie theory, a one-dimensional subspace of a Lie algebra is mapped to a one-parameter group under the Lie group–Lie algebra correspondence
One-dimensional_space
Algebraic structure in mathematics
enveloping algebra of a Lie algebra, with generators a basis of the Lie algebra and relations of the form XY – YX – [X, Y] = 0. The tensor algebra, symmetric
Quadratic_algebra
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COMPLEX LIE-ALGEBRA
COMPLEX LIE-ALGEBRA
COMPLEX LIE-ALGEBRA
COMPLEX LIE-ALGEBRA
COMPLEX LIE-ALGEBRA
COMPLEX LIE-ALGEBRA
COMPLEX LIE-ALGEBRA
COMPLEX LIE-ALGEBRA
COMPLEX LIE-ALGEBRA
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