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;...

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...

{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...

collaborative software Loop (algebra), a quasigroup with an identity element Loop (graph theory), an edge that begins and ends on the same vertex Loop (topology)...

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

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...

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...

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

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...

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...

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...

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

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

a Lie algebra g {\displaystyle {\mathfrak {g}}} is solvable if its derived series terminates in the zero subalgebra. The derived Lie algebra of the Lie...

holes both in string theory and in other approaches to quantum gravity like loop quantum gravity. Unsolved problem in physics Is physical information lost...

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

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...

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...

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...

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

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...

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