affine Lie

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  • Therefore the central extensions of an affine Lie group are classified by a single parameter k which is called the level in the physics literature, where it first appeared.
  • This topological obstruction can also be seen in the representation theory of the affine Lie algebra symmetry of the theory.
  • The answer is that one gets sums of finite-dimensional and affine Lie algebras.
  • When each level is a positive integer the affine Lie algebra has unitary highest weight representations with highest weights that are dominant integral.
  • An affine Lie algebra can always be constructed as a central extension of the loop algebra of the corresponding simple Lie algebra.
  • The point of view of string theory helps to understand many deep properties of affine Lie algebras, such as the fact that the characters of their representations transform amongst themselves under the modular group.
  • Their central extensions are precisely the twisted affine Lie algebras.
  • In particular, this group always contains the identity element, and the corresponding affine Lie algebra is called an untwisted affine Lie algebra.
  • An affine Lie algebra is a particular type of Kac-Moody algebra.
  • When the simple algebra admits automorphisms that are not inner automorphisms, one may obtain other Dynkin diagrams and these correspond to twisted affine Lie algebras.
  • Central extensions of the affine Lie group by a single generator are topologically circle bundles over this free loop group, which are classified by a two-class known as the first Chern class of the fibration.
  • In 2013 he became a fellow of the American Mathematical Society, for "contributions to representation theory, conformal field theory, affine Lie algebras, and quantum field theory".
  • 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.
  • Affine Lie algebras are a special case of Kac-Moody algebras, which have particular importance in mathematics and theoretical physics, especially conformal field theory and the theory of exactly solvable models.
  • They are used in the classification of affine Lie algebras and superalgebras, and semisimple p-adic algebraic groups, and correspond to families of Macdonald polynomials.
  • The Dynkin diagram of each affine Lie algebra consists of that of the corresponding simple Lie algebra plus an additional node, which corresponds to the addition of an imaginary root.
  • The affine Lie algebra corresponding to a finite-dimensional semisimple Lie algebra is the direct sum of the affine Lie algebras corresponding to its simple summands.
  • From purely mathematical point of view, affine Lie algebras are interesting because their representation theory, like representation theory of finite dimensional, semisimple Lie algebras is much better understood than that of general Kac-Moody algebras.
  • As observed by Victor Kac, the character formula for representations of affine Lie algebras implies certain combinatorial identities, the Macdonald identities.
  • Young's lattice prominently figures in algebraic combinatorics, forming the simplest example of a differential poset in the sense of, It is also closely connected with the crystal bases for affine Lie algebras.
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