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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).

Applications, Index theory & Central extensions and representation theory

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Overview

Definition

Basic constructions

Infinite-dimensional Lie group structure

Homogeneous spaces and factorization

Central extensions and representation theory

Twisted loop groups

Algebraic loop groups

Complex and holomorphic loop groups

Examples

Index theory

Applications

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Loop group

Nodes66
Edges65
Triples123
Avg. degree1.97
Density0.030303
Components1

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Loop group

Top relations

related to References · 34
Loop group → Advances, American Mathematical Society, Andrew, Annals, Bibcode, Chuu-Lian, Clarendon Press, Constantin, Daniel, Differential Geometry, Freed, Geometry, George, Graeme, Hopkins, ISBN, Journal, K-theory III, Karen, Lecture Notes
related to Index theory · 13
Loop group → Analytic, Dirac, Freed, Grassmannian, Hopkins, In, K-theory, Loop, One, Pressley, Segal, Teleman, Toeplitz
related to Integrable systems · 13
Loop group → Baker, Grassmannian, In, KdV, Laurent-polynomially, Lax, Lie, Loop, L𝔤, Segal, Splittings, When, Wilson
related to Central extensions and representation theory · 10
Loop group → BG, For, H4, Kac, LG, Lie, Many, Moody, Such, The
related to Homogeneous spaces and factorization · 10
Loop group → Birkhoff, Bruhat, Gℂ, If, LG/G, LGℂ, The, These, This, Toeplitz
has application · 9
Loop group → Chern, Grassmannians, H-space, In, K-theory, Kac, Loop, Moody, Simons
related to Complex and holomorphic loop groups · 5
Loop group → GC, If, LG, Lie, This
related to Examples · 5
Loop group → In, Lie, More, S1, The
related to Hodge theory · 5
Loop group → From, Hodge, In, Jeremy Daniel, Loop
related to Algebraic loop groups · 4
Loop group → If, In, Laurent, The

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Important terminology

loop group groups theory lie lg one infinite-dimensional smooth compact loops representations representation central geometry s1 affine topology twisted based

Entity relationships Subject–Predicate–Object triples

SubjectPredicateObjectConfidenceSrc
Loop groupis afunctor L G0.90text
constant-mean-curvatureinstance ofand a range of geometric integrable systems0.80text
isothermic surfacesinstance ofand a range of geometric integrable systems0.80text
Loop grouphas applicationLoop0.60section
Loop grouphas applicationIn0.60section
Loop grouphas applicationH-space0.60section
Loop grouphas applicationKac0.60section
Loop grouphas applicationMoody0.60section
Loop grouphas applicationGrassmannians0.60section
Loop grouphas applicationChern0.60section
Loop grouphas applicationSimons0.60section
Loop grouphas applicationK-theory0.60section

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