Lesson plan
Loop group
In mathematics, a loop group is, in the most common Lie-theoretic sense, the group of smooth maps from the circle to a Lie group , with multiplication defined pointwise. When is a compact Lie group, is a basic example of an infinite-dimensional Lie group, with Lie algebra . The subgroup of based loops is fundamental in homotopy theory, while central extensions of loop groups and their projective representations are closely related to affine Kac–Moody algebras, conformal field theory, and the Verlinde formula. In algebraic geometry one also studies algebraic loop groups, defined by , together with their associated affine Grassmannians and affine flag varieties.
Article summary
In mathematics, a loop group is, in the most common Lie-theoretic sense, the group of smooth maps from the circle to a Lie group , with multiplication defined pointwise. When is a compact Lie group, is a basic example of an infinite-dimensional Lie group, with Lie algebra . The subgroup of based loops is fundamental in homotopy theory, while central extensions of loop groups and their projective representations are closely related to affine Kac–Moody algebras, conformal field theory, and the Verlinde formula. In algebraic geometry one also studies algebraic loop groups, defined by , together with their associated affine Grassmannians and affine flag varieties.
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