Loop group

In mathematics, a loop group is, in the most common Lie-theoretic sense, the group LG = C^∞(S¹, G) of smooth maps from the circle S¹ to a Lie group G, with multiplication defined pointwise. When G is a compact Lie group, LG is a basic example of an infinite-dimensional Lie group, with Lie algebra L𝔤 = C^∞(S¹, 𝔤).

The subgroup ΩG 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 LG(R) = G(R((t))), together with their associated affine Grassmannians and affine flag varieties.