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Algebraic character

An algebraic character is a formal expression attached to a module in representation theory of semisimple Lie algebras that generalizes the character of a finite-dimensional representation and is anal

Lie algebroid

In mathematics, a Lie algebroid is a vector bundle together with a Lie bracket on its space of sections and a vector bundle morphism , satisfying a Leibniz rule. A Lie algebroid can thus be thought of

Gelfand–Fuks cohomology

In mathematics, Gelfand–Fuks cohomology, introduced in, is a cohomology theory for Lie algebras of smooth vector fields. It differs from the Lie algebra cohomology of Chevalley-Eilenberg in that its c

Lie algebra cohomology

In 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 and homogeneous spaces by relating co

Manin triple

In mathematics, a Manin triple (g, p, q) consists of a Lie algebra g with a non-degenerate invariant symmetric bilinear form, together with two isotropic subalgebras p and q such that g is the direct

Group contraction

In theoretical physics, Eugene Wigner and Erdal İnönü have discussed the possibility to obtain from a given Lie group a different (non-isomorphic) Lie group by a group contraction with respect to a co

Simple Lie algebra

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 is one of the major achievements of Wilhelm

Restricted root system

In mathematics, restricted root systems, sometimes called relative root systems, are the root systems associated with a symmetric space. The associated finite reflection group is called the restricted

Argument shift method

In mathematics, the argument shift method is a method for constructing functions in involution with respect to Poisson–Lie brackets, introduced by Mishchenko and Fomenko. They used it to prove that th

Index of a Lie algebra

In algebra, let g be a Lie algebra over a field K. Let further be a one-form on g. The stabilizer gξ of ξ is the Lie subalgebra of elements of g that annihilate ξ in the coadjoint representation. The

Linear Lie algebra

In algebra, a linear Lie algebra is a subalgebra of the Lie algebra consisting of endomorphisms of a vector space V. In other words, a linear Lie algebra is the image of a Lie algebra representation.

Capelli's identity

In mathematics, Capelli's identity, named after Alfredo Capelli, is an analogue of the formula det(AB) = det(A) det(B), for certain matrices with noncommuting entries, related to the representation th

Exponential map (Lie theory)

In the theory of Lie groups, the exponential map is a map from the Lie algebra of a Lie group to the group, which allows one to recapture the local group structure from the Lie algebra. The existence

Particle physics and representation theory

There is a natural connection between particle physics and representation theory, as first noted in the 1930s by Eugene Wigner. It links the properties of elementary particles to the structure of Lie

Levi decomposition

In Lie theory and representation theory, the Levi decomposition, conjectured by Wilhelm Killing and Élie Cartan and proved by Eugenio Elia Levi, states that any finite-dimensional real Lie algebra g i

Modular Lie algebra

In mathematics, a modular Lie algebra is a Lie algebra over a field of positive characteristic. The theory of modular Lie algebras is significantly different from the theory of real and complex Lie al

Radical of a Lie algebra

In the mathematical field of Lie theory, the radical of a Lie algebra is the largest solvable ideal of The radical, denoted by , fits into the exact sequence . where is semisimple. When the ground fie

Anyonic Lie algebra

In mathematics, an anyonic Lie algebra is a U(1) graded vector space over equipped with a bilinear operator and linear maps (some authors use ) and such that , satisfying following axioms:
*
*
*
*

Littelmann path model

In mathematics, the Littelmann path model is a combinatorial device due to Peter Littelmann for computing multiplicities without overcounting in the representation theory of symmetrisable Kac–Moody al

Canonical basis

In mathematics, a canonical basis is a basis of an algebraic structure that is canonical in a sense that depends on the precise context:
* In a coordinate space, and more generally in a free module,

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, esp

Simplicial Lie algebra

In algebra, a simplicial Lie algebra is a simplicial object in the category of Lie algebras. In particular, it is a simplicial abelian group, and thus is subject to the Dold–Kan correspondence.

Supermathematics

Supermathematics is the branch of mathematical physics which applies the mathematics of Lie superalgebras to the behaviour of bosons and fermions. The driving force in its formation in the 1960s and 1

Lie coalgebra

In 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 has the structure of a Lie coalgeb

Vogel plane

In mathematics, the Vogel plane is a method of parameterizing simple Lie algebras by eigenvalues α, β, γ of the Casimir operator on the symmetric square of the Lie algebra, which gives a point (α: β:

Special orthogonal Lie algebra

No description available.

Symplectic Lie algebra

No description available.

Cartan matrix

In mathematics, the term Cartan matrix has three meanings. All of these are named after the French mathematician Élie Cartan. Amusingly, the Cartan matrices in the context of Lie algebras were first i

Pre-Lie algebra

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 fields on affine space. The notion of pre-Lie al

Lie group–Lie algebra correspondence

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 such a relationship. Lie groups that are isomorph

Quasi-Lie algebra

In mathematics, a quasi-Lie algebra in abstract algebra is just like a Lie algebra, but with the usual axiom replaced by (anti-symmetry). In characteristic other than 2, these are equivalent (in the p

Malcev Lie 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. Both were introduced by , Appendix A3), based on

Supersymmetry algebras in 1 + 1 dimensions

A two dimensional Minkowski space, i.e. a flat space with one time and one spatial dimension, has a two-dimensional Poincaré group IO(1,1) as its symmetry group. The respective Lie algebra is called t

Weyl 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 of that root system. Specifically, it is the

Duflo isomorphism

In mathematics, the Duflo isomorphism is an isomorphism between the center of the universal enveloping algebra of a finite-dimensional Lie algebra and the invariants of its symmetric algebra. It was i

Lie conformal 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 algebras are very closely rela

Lie algebra

In mathematics, a Lie algebra (pronounced /liː/ LEE) is a vector space together with an operation called the Lie bracket, an alternating bilinear map , that satisfies the Jacobi identity. The Lie brac

Weyl's theorem on complete reducibility

In algebra, Weyl's theorem on complete reducibility is a fundamental result in the theory of Lie algebra representations (specifically in the representation theory of semisimple Lie algebras). Let be

Folding (Dynkin diagram)

No description available.

Cartan pair

In the mathematical fields of Lie theory and algebraic topology, the notion of Cartan pair is a technical condition on the relationship between a reductive Lie algebra and a subalgebra reductive in .

Quillen's lemma

In algebra, Quillen's lemma states that an endomorphism of a simple module over the enveloping algebra of a finite-dimensional Lie algebra over a field k is algebraic over k. In contrast to a version

Differential graded Lie algebra

In mathematics, in particular 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 complex structures

Vertex operator algebra

In mathematics, a vertex operator algebra (VOA) is an algebraic structure that plays an important role in two-dimensional conformal field theory and string theory. In addition to physical applications

Knizhnik–Zamolodchikov equations

In mathematical physics the Knizhnik–Zamolodchikov equations, or KZ equations, are linear differential equations satisfied by the correlation functions (on the Riemann sphere) of two-dimensional confo

Whitehead's lemma (Lie algebra)

In homological algebra, Whitehead's lemmas (named after J. H. C. Whitehead) represent a series of statements regarding representation theory of finite-dimensional, semisimple Lie algebras in character

Jordan–Chevalley decomposition

In mathematics, the Jordan–Chevalley decomposition, named after Camille Jordan and Claude Chevalley, expresses a linear operator as the sum of its commuting semisimple part and its nilpotent part. The

Nilpotent cone

In mathematics, the nilpotent cone of a finite-dimensional semisimple Lie algebra is the set of elements that act nilpotently in all representations of In other words, The nilpotent cone is an irreduc

Generalized Kac–Moody algebra

In mathematics, a generalized Kac–Moody algebra is a Lie algebra that is similar to a Kac–Moody algebra, except that it is allowed to have imaginary simple roots. Generalized Kac–Moody algebras are al

Bianchi classification

In mathematics, the Bianchi classification provides a list of all real 3-dimensional Lie algebras (up to isomorphism). The classification contains 11 classes, 9 of which contain a single Lie algebra a

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 a

Toda field theory

In mathematics and physics, specifically the study of field theory and partial differential equations, a Toda field theory, named after Morikazu Toda, is specified by a choice of Kac–Moody algebra and

Nilradical of a Lie algebra

In algebra, the nilradical of a Lie algebra is a nilpotent ideal, which is as large as possible. The nilradical of a finite-dimensional Lie algebra is its maximal nilpotent ideal, which exists because

Tate vector space

In mathematics, a Tate vector space is a vector space obtained from finite-dimensional vector spaces in a way that makes it possible to extend concepts such as dimension and determinant to an infinite

Lie–Palais theorem

In differential geometry, the Lie–Palais theorem states that an action of a finite-dimensional Lie algebra on a smooth compact manifold can be lifted to an action of a finite-dimensional Lie group. Fo

Engel subalgebra

In mathematics, an Engel subalgebra of a Lie algebra with respect to some element x is the subalgebra of elements annihilated by some power of ad x. Engel subalgebras are named after Friedrich Engel.

Whitehead product

In mathematics, the Whitehead product is a graded quasi-Lie algebra structure on the homotopy groups of a space. It was defined by J. H. C. Whitehead in. The relevant MSC code is: 55Q15, Whitehead pro

Engel group

In mathematics, an element x of a Lie group or a Lie algebra is called an n-Engel element, named after Friedrich Engel, if it satisfies the n-Engel condition that the repeated commutator [...[[x,y],y]

Quadratic Lie algebra

A quadratic Lie algebra is a Lie algebra together with a compatible symmetric bilinear form. Compatibility means that it is invariant under the adjoint representation. Examples of such are semisimple

Regular element of a Lie algebra

In mathematics, a regular element 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 element is regul

Jacobson–Morozov theorem

In mathematics, the Jacobson–Morozov theorem is the assertion that nilpotent elements in a semi-simple Lie algebra can be extended to sl2-triples. The theorem is named after , .

Chiral Lie algebra

In algebra, a chiral Lie algebra is a D-module on a curve with a certain structure of Lie algebra. It is related to an -algebra via the Riemann–Hilbert correspondence.

Kac–Moody algebra

In mathematics, 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, that can be de

Lie–Kolchin theorem

In mathematics, the Lie–Kolchin theorem is a theorem in the representation theory of linear algebraic groups; Lie's theorem is the analog for linear Lie algebras. It states that if G is a connected an

Lie algebra extension

In the theory of 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 h. Extensions arise in several wa

Magnus expansion

In mathematics and physics, the Magnus expansion, named after Wilhelm Magnus (1907–1990), provides an exponential representation of the solution of a first-order homogeneous linear differential equati

Orthogonal symmetric Lie algebra

In mathematics, an orthogonal symmetric Lie algebra is a pair consisting of a real Lie algebra and an automorphism of of order such that the eigenspace of s corresponding to 1 (i.e., the set of fixed

Cartan subalgebra

In mathematics, a Cartan subalgebra, often abbreviated as CSA, is a nilpotent subalgebra of a Lie algebra that is self-normalising (if for all , then ). They were introduced by Élie Cartan in his doct

Lie bialgebra

In mathematics, a Lie bialgebra is the Lie-theoretic case of a bialgebra: it is a set with a Lie algebra and a Lie coalgebra structure which are compatible. It is a bialgebra where the multiplication

N = 2 superconformal algebra

In mathematical physics, the 2D N = 2 superconformal algebra is an infinite-dimensional Lie superalgebra, related to supersymmetry, that occurs in string theory and two-dimensional conformal field the

Loop algebra

In mathematics, loop algebras are certain types of Lie algebras, of particular interest in theoretical physics.

Coset construction

In mathematics, the coset construction (or GKO construction) is a method of constructing unitary highest weight representations of the Virasoro algebra, introduced by Peter Goddard, Adrian Kent and Da

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

List of Lie groups topics

This is a list of Lie group topics, by Wikipedia page.

Beilinson–Bernstein localization

In mathematics, especially in representation theory and algebraic geometry, the Beilinson–Bernstein localization theorem relates D-modules on flag varieties G/B to representations of the Lie algebra a

Super-Poincaré algebra

In theoretical physics, a super-Poincaré algebra is an extension of the Poincaré algebra to incorporate supersymmetry, a relation between bosons and fermions. They are examples of supersymmetry algebr

Affine root system

In mathematics, an affine root system is a root system of affine-linear functions on a Euclidean space. They are used in the classification of affine Lie algebras and superalgebras, and semisimple p-a

Kantor–Koecher–Tits construction

In algebra, the Kantor–Koecher–Tits construction is a method of constructing a Lie algebra from a Jordan algebra, introduced by Jacques Tits, Kantor, and Koecher. If J is a Jordan algebra, the Kantor–

Solvmanifold

In mathematics, a solvmanifold is a homogeneous space of a connected solvable Lie group. It may also be characterized as a quotient of a connected solvable Lie group by a closed subgroup. (Some author

Hypoalgebra

In algebra, a hypoalgebra is a generalization of a subalgebra of a Lie algebra introduced by . The relation between an algebra and a hypoalgebra is called a subjoining, which generalizes the notion of

Crystal base

A crystal base for a representation of a quantum group on a -vector spaceis not a base of that vector space but rather a -base of where is a -lattice in that vector spaces. Crystal bases appeared in t

Exceptional 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 five of them: ; their respective dimensions are

Glossary of Lie groups and Lie algebras

This is a glossary for the terminology applied in the mathematical theories of Lie groups and Lie algebras. For the topics in the representation theory of Lie groups and Lie algebras, see Glossary of

R-algebroid

In mathematics, R-algebroids are constructed starting from groupoids. These are more abstract concepts than the Lie algebroids that play a similar role in the theory of Lie groupoids to that of Lie al

Restricted Lie algebra

In mathematics, a restricted Lie algebra is a Lie algebra together with an additional "p operation."

Satake diagram

In the mathematical study of Lie algebras and Lie groups, a Satake diagram is a generalization of a Dynkin diagram introduced by Satake whose configurations classify simple Lie algebras over the field

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, or equivalently, a one-dimensional representation of A over F. It is

Splitting Cartan subalgebra

No description available.

Cartan's criterion

In mathematics, Cartan's criterion 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 based on the not

Simple Lie group

In mathematics, a simple Lie group is a connected non-abelian Lie group G which does not have nontrivial connected normal subgroups. The list of simple Lie groups can be used to read off the list of s

Cartan decomposition

In mathematics, the Cartan decomposition is a decomposition of a semisimple Lie group or Lie algebra, which plays an important role in their structure theory and representation theory. It generalizes

Sl2-triple

In the theory of Lie algebras, an sl2-triple is a triple of elements of a Lie algebra that satisfy the commutation relations between the standard generators of the special linear Lie algebra sl2. This

Lie-* algebra

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 the conformal algebras discussed by and to ve

Current algebra

Certain commutation relations among the current density operators in quantum field theories define an infinite-dimensional Lie algebra called a current algebra. Mathematically these are Lie algebras c

Generalised Whitehead product

The Whitehead product is a mathematical construction introduced in . It has been a useful tool in determining the properties of spaces. The mathematical notion of space includes every shape that exist

Invariant convex cone

In mathematics, an invariant convex cone is a closed convex cone in a Lie algebra of a connected Lie group that is invariant under inner automorphisms. The study of such cones was initiated by Ernest

Jantzen filtration

In representation theory, a Jantzen filtration is a filtration of a Verma module of a semisimple Lie algebra, or a Weyl module of a reductive algebraic group of positive characteristic. Jantzen filtra

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 nat

Springer resolution

In mathematics, the Springer resolution is a resolution of the variety of nilpotent elements in a semisimple Lie algebra, or the unipotent elements of a reductive algebraic group, introduced by Tonny

Monster Lie 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 the monstrous moonshine conjectures.

Supersymmetry algebra

In theoretical physics, a supersymmetry algebra (or SUSY algebra) is a mathematical formalism for describing the relation between bosons and fermions. The supersymmetry algebra contains not only the P

Ringel–Hall algebra

In mathematics, a Ringel–Hall algebra is a generalization of the Hall algebra, studied by Claus Michael Ringel. It has a basis of equivalence classes of objects of an abelian category, and the structu

Adjoint bundle

In mathematics, an adjoint bundle is a vector bundle naturally associated to any principal bundle. The fibers of the adjoint bundle carry a Lie algebra structure making the adjoint bundle into a (nona

Chevalley basis

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 are integers. Chevalley used these bases to

Ado's theorem

In abstract algebra, Ado's theorem is a theorem characterizing finite-dimensional Lie algebras.

Dixmier mapping

In mathematics, the Dixmier mapping describes the space Prim(U(g)) of primitive ideals of the universal enveloping algebra U(g) of a finite-dimensional solvable Lie algebra g over an algebraically clo

Parabolic Lie algebra

In algebra, a parabolic Lie algebra is a subalgebra of a semisimple Lie algebra satisfying one of the following two conditions:
* contains a maximal solvable subalgebra (a Borel subalgebra) of ;
* t

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 nonassoci

Jacobi identity

In mathematics, the Jacobi identity is a property of a binary operation that describes how the order of evaluation, the placement of parentheses in a multiple product, affects the result of the operat

Centralizer and normalizer

In mathematics, especially group theory, the centralizer (also called commutant) of a subset S in a group G is the set of elements of G such that each member commutes with each element of S, or equiva

Homotopy Lie algebra

In mathematics, in particular abstract algebra and topology, a homotopy Lie algebra (or -algebra) is a generalisation of the concept of a differential graded Lie algebra. To be a little more specific,

(B, N) pair

In mathematics, a (B, N) pair is a structure on groups of Lie type that allows one to give uniform proofs of many results, instead of giving a large number of case-by-case proofs. Roughly speaking, it

Classification of low-dimensional real Lie algebras

This mathematics-related list provides Mubarakzyanov's classification of low-dimensional real Lie algebras, published in Russian in 1963. It complements the article on Lie algebra in the area of abstr

Lie's third theorem

In the mathematics of Lie theory, Lie's third theorem states that every finite-dimensional Lie algebra over the real numbers is associated to a Lie group . The theorem is part of the Lie group–Lie alg

Symmetric cone

In mathematics, symmetric cones, sometimes called domains of positivity, are open convex self-dual cones in Euclidean space which have a transitive group of symmetries, i.e. invertible operators that

Chevalley restriction theorem

In the mathematical theory of Lie groups, the Chevalley restriction theorem describes functions on a Lie algebra which are invariant under the action of a Lie group in terms of functions on a Cartan s

Dynkin diagram

In the mathematical field of Lie theory, a Dynkin diagram, named for Eugene Dynkin, is a type of graph with some edges doubled or tripled (drawn as a double or triple line). Dynkin diagrams arise in t

Gell-Mann matrices

The Gell-Mann matrices, developed by Murray Gell-Mann, are a set of eight linearly independent 3×3 traceless Hermitian matrices used in the study of the strong interaction in particle physics.They spa

Theorem of the highest weight

In representation theory, a branch of mathematics, the theorem of the highest weight classifies the irreducible representations of a complex semisimple Lie algebra . There is a closely related theorem

Weil algebra

In mathematics, the Weil algebra of a Lie algebra g, introduced by Cartan based on unpublished work of André Weil, is a differential graded algebra given by the Koszul algebra Λ(g*)⊗S(g*) of its dual

Lie algebra-valued differential form

In differential geometry, a Lie-algebra-valued form is a differential form with values in a Lie algebra. Such forms have important applications in the theory of connections on a principal bundle as we

Representation theory of semisimple Lie algebras

In mathematics, the 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 by E. Cartan and

Automorphism of a Lie algebra

In abstract algebra, an automorphism of a Lie algebra is an isomorphism between and itself; i.e., a linear automorphism that preserves the bracket. The totality of them forms the automorphism group of

Macdonald identities

In mathematics, the Macdonald identities are some infinite product identities associated to affine root systems, introduced by Ian Macdonald. They include as special cases the Jacobi triple product id

Borel subalgebra

In mathematics, specifically in representation theory, a Borel subalgebra of a Lie algebra is a maximal solvable subalgebra. The notion is named after Armand Borel. If the Lie algebra is the Lie algeb

Principal subalgebra

In mathematics, a principal subalgebra of a complex simple Lie algebra is a 3-dimensional simple subalgebra whose non-zero elements are regular. A finite-dimensional complex simple Lie algebra has a u

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

Borel–de Siebenthal theory

In mathematics, Borel–de Siebenthal theory describes the closed connected subgroups of a compact Lie group that have maximal rank, i.e. contain a maximal torus. It is named after the Swiss mathematici

Leibniz algebra

In mathematics, a (right) Leibniz algebra, named after Gottfried Wilhelm Leibniz, sometimes called a Loday algebra, after Jean-Louis Loday, is a module L over a commutative ring R with a bilinear prod

Atiyah algebroid

In mathematics, the Atiyah algebroid, or Atiyah sequence, of a principal -bundle over a manifold , where is a Lie group, is the Lie algebroid of the gauge groupoid of . Explicitly, it is given by the

Superconformal algebra

In theoretical physics, the superconformal algebra is a graded Lie algebra or superalgebra that combines the conformal algebra and supersymmetry. In two dimensions, the superconformal algebra is infin

Group of Lie type

In mathematics, specifically in group theory, the phrase group of Lie type usually refers to finite groups that are closely related to the group of rational points of a reductive linear algebraic grou

Zinbiel algebra

In mathematics, a Zinbiel algebra or dual Leibniz algebra is a module over a commutative ring with a bilinear product satisfying the defining identity: Zinbiel algebras were introduced by Jean-Louis L

Malcev algebra

In mathematics, a Malcev algebra (or Maltsev algebra or Moufang–Lie algebra) over a field is a nonassociative algebra that is antisymmetric, so that and satisfies the Malcev identity They were first d

Takiff algebra

In mathematics, a Takiff algebra is a Lie algebra over a truncated polynomial ring. More precisely, a Takiff algebra of a Lie algebra g over a field k is a Lie algebra of the form g[x]/(xn+1) = g⊗kk[x

Bivector (complex)

In mathematics, a bivector is the vector part of a biquaternion. For biquaternion q = w + xi + yj + zk, w is called the biscalar and xi + yj + zk is its bivector part. The coordinates w, x, y, z are c

Moufang polygon

In mathematics, Moufang polygons are a generalization by Jacques Tits of the Moufang planes studied by Ruth Moufang, and are irreducible buildings of rank two that admit the action of .In a book on th

Group analysis of differential equations

Group analysis of differential equations is a branch of mathematics that studies the symmetry properties of differential equations with respect to various transformations of independent and dependent

Quasi-Frobenius Lie algebra

In mathematics, a quasi-Frobenius Lie algebra over a field is a Lie algebra equipped with a nondegenerate skew-symmetric bilinear form , which is a Lie algebra 2- of with values in . In other words, f

Super Virasoro algebra

In mathematical physics, a super Virasoro algebra is an extension of the Virasoro algebra (named after Miguel Ángel Virasoro) to a Lie superalgebra. There are two extensions with particular importance

Nilpotent orbit

In mathematics, nilpotent orbits are generalizations of nilpotent matrices that play an important rolein representation theory of real and complex semisimple Lie groups and semisimple Lie algebras.

Structure constants

In mathematics, the structure constants or structure coefficients of an algebra over a field are used to explicitly specify the product of two basis vectors in the algebra as a linear combination. Giv

Virasoro algebra

In mathematics, the Virasoro algebra (named after the physicist Miguel Ángel Virasoro) is a complex Lie algebra and the unique central extension of the Witt algebra. It is widely used in two-dimension

Ring of modular forms

In mathematics, the ring of modular forms associated to a subgroup Γ of the special linear group SL(2, Z) is the graded ring generated by the modular forms of Γ. The study of rings of modular forms de

Vogan diagram

In mathematics, a Vogan diagram, named after David Vogan, is a variation of the Dynkin diagram of a real semisimple Lie algebra that indicates the maximal compact subgroup. Although they resemble Sata

Clebsch–Gordan coefficients for SU(3)

In mathematical physics, Clebsch–Gordan coefficients are the expansion coefficients of total angular momentum eigenstates in an uncoupled tensor product basis. Mathematically, they specify the decompo

Nijenhuis–Richardson bracket

In mathematics, the algebraic bracket or Nijenhuis–Richardson bracket is a graded Lie algebra structure on the space of alternating multilinear forms of a vector space to itself, introduced by A. Nije

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 complex

Complexification (Lie group)

In mathematics, the complexification or universal complexification of a real Lie group is given by a continuous homomorphism of the group into a complex Lie group with the universal property that ever

Casimir element

In mathematics, a Casimir element (also known as a Casimir invariant or Casimir operator) is a distinguished element of the center of the universal enveloping algebra of a Lie algebra. A prototypical

Special linear Lie algebra

In mathematics, the special linear Lie algebra of order n (denoted or ) is the Lie algebra of matrices with trace zero and with the Lie bracket . This algebra is well studied and understood, and is of

Valya algebra

In abstract algebra, a Valya algebra (or Valentina algebra) is a nonassociative algebra M over a field F whose multiplicative binary operation g satisfies the following axioms: 1. The skew-symmetry co

Kostant's convexity theorem

In mathematics, Kostant's convexity theorem, introduced by Bertram Kostant, states that the projection of every coadjoint orbit of a connected compact Lie group into the dual of a Cartan subalgebra is

Poincaré–Birkhoff–Witt theorem

In mathematics, more specifically in the theory of Lie algebras, the Poincaré–Birkhoff–Witt theorem (or PBW theorem) is a result giving an explicit description of the universal enveloping algebra of a

Engel identity

The Engel identity, named after Friedrich Engel, is a mathematical equation that is satisfied by all elements of a Lie ring, in the case of an Engel Lie ring, or by all the elements of a group, in the

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 solv

Serre's theorem on a semisimple Lie algebra

In abstract algebra, specifically the theory of Lie algebras, Serre's theorem states: given a (finite reduced) root system , there exists a finite-dimensional semisimple Lie algebra whose root system

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