In mathematics, loop algebras are certain types of Lie algebras, of particular interest in theoretical physics. For a Lie algebra g {\displaystyle {\mathfrak...

In mathematics, especially in abstract algebra, a quasigroup is an algebraic structure that resembles a group in the sense that "division" is always possible...

constructed: starting from a simple Lie algebra g {\displaystyle {\mathfrak {g}}} , one considers the loop algebra, L g {\displaystyle L{\mathfrak {g}}}...

E., eds. (1997), "Representations of loop algebras", Lie Algebras - Finite and Infinite Dimensional Lie Algebras and Applications in Physics, Studies...

algebra which is isomorphic with an untwisted affine Kac–Moody algebra. Using the centrally extended loop algebra one may construct a current algebra...

0 with 1. The set of all loops in X forms a space called the loop space of X. Free loop Loop group Loop space Loop algebra Fundamental group Quasigroup...

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...

loops have an associated algebra, the Malcev algebra, similar in some ways to how a Lie group has an associated Lie algebra. A Moufang loop is a loop...

of a Hilbert space and associated operators reproducing the correct loop algebra) have been given by two groups (Lewandowski, Okołów, Sahlmann and Thiemann;...

{Spin} (1,3)} . The Poincaré algebra is the Lie algebra of the Poincaré group. It is a Lie algebra extension of the Lie algebra of the Lorentz group. More...

iteration of the loop containing it Microsoft Loop, collaborative software Loop (algebra), a quasigroup with an identity element Loop (graph theory), an...

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...

In mathematics, a Lie algebra (pronounced /liː/ LEE) is a vector space g {\displaystyle {\mathfrak {g}}} together with an operation called the Lie bracket...

current algebras arise naturally as a central extension of the loop algebra, known as Kac–Moody algebras or, more specifically, affine Lie algebras. In this...

This (real) Lie algebra has dimension n2 − 1. More information about the structure of this Lie algebra can be found below in § Lie algebra structure. In...

mathematician Élie Cartan. Amusingly, the Cartan matrices in the context of Lie algebras were first investigated by Wilhelm Killing, whereas the Killing form is...

is a nilpotent subalgebra h {\displaystyle {\mathfrak {h}}} of a Lie algebra g {\displaystyle {\mathfrak {g}}} that is self-normalising (if [ X , Y...

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 0...

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...

Representation theory of Hopf algebras General Associative property, Associator Heap (mathematics) Magma (algebra) Loop (algebra), Quasigroup Nonassociative...

In algebraic and differential geometry, a Calabi–Yau manifold, also known as a Calabi–Yau space, is a particular type of manifold which has certain properties...

the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if G is G L ( n , R ) {\displaystyle...

form and a split real form), their Lie algebras g 2 , {\displaystyle {\mathfrak {g}}_{2},} as well as some algebraic groups. They are the smallest of the...

several closely related exceptional simple Lie groups, linear algebraic groups or Lie algebras of dimension 248; the same notation is used for the corresponding...

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...

Diffeomorphism Loop Euclidean Lie algebras Lie group–Lie algebra correspondence Exponential map Adjoint representation Killing form Index Simple Lie algebra Loop 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...

group. The unitary group U(n) is a real Lie group of dimension n2. The Lie algebra of U(n) consists of n × n skew-Hermitian matrices, with the Lie bracket...

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)...

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...