Algebraic Notion
* Idea: A generalization of a finite group, in which the binary operation is not associative.
* Examples: The smallest loops that are not themselves groups are those of order five.
@ And physics: Frampton et al ht/01-fs.

Topological Notion
$Def: An equivalence class of closed curves on a manifold, where two are equivalent if they differ by retraced segments.$ Hoop: An equivalence class of loops, where two differ if they have the same holonomy for all connections in a given fiber bundle.
* Small loop: A loop which is homotopic to a loop contained in an arbitrarily small neighborhood of its base point.
@ Dynamics: Kondev PRL(97) [field theory of fluctuating loops]; Arreaga et al PRE(02)cm/01 [equilibrium configurations with constraints].
@ Generalizations: Griego gq/95 [and applications to knot theory and quantum gravity].
> Related topics: see Cuntz Algebra; knot theory; Link Theory; tiling.

Loop Group
$Def: For a given manifold, it is the set of loops based at a point p in M, with the natural, non-commutative composition α $$\circ$$ β:= α followed by β. * Topology: It is a topological group, with either (i) α in Uε(β) if there exist curves a in α and b in β, with a in Uε(b) in the usual sense of curves; or (ii) based on holonomies [@ Barrett IJTP(91)]. * Loop algebra: The Lie algebra of a loop group.$ For a given group: The group of maps f : S1G from the circle to a fixed finite-dimensional group G, with composition law (fg)(s):= f(s) g(s).
@ General references: Adams 78; Pressley & Segal 86; Bars NPB(89); Rasmussen & Weis ht/94 [hoop group topology]; Solomon a1303 [comment].
@ Representations: Carey & Langmann in(02)-a1007 [survey, and quantum field theory].
@ Generalizations: Di Bartolo et al CMP(93)gq, PRD(95)gq/94 [extended loop group]; Leal PRD(02)ht [signed points].
@ Related topics: Spallanzani CMP(01) [relationship with hoops]; Mickelsson in(06)mp/04 [central extension]; Frenkel & Zhu a0810 [double loop groups, gerbal representations]; Zeitlin JFA(12)-a1012 [loop ax+b group, unitary representations]; Carpi & Hillier RVMP(17)-a1509 [and non-commutative geometry].

Loop Space
@ General references: Adams 78; Bars NPB(89); Morozov et al PLB(91) [loop space geometry and supersymmetry]; Lempert JDG(93).
@ Calculus: Cattaneo et al CMP(99) [connections]; Reiris & Spallanzani CQG(99) [loop derivative]; Pickrell mp/04 [invariant measure]; Reyes JMP(07)ht/06 [operators on loop functions]; > s.a. Paths.
@ Related topics: Wurzbacher JGP(95) [symplectic geometry]; Sergeev TMP(08) [compact Lie group, twistor quantization].

Loop-Related Physical Systems
@ General references: Rajeev ht/04-conf [Yang-Mills theory and loop space]; Ferreira & Luchini NPB(12)-a1109 [and the generalized non-abelian Stokes theorems for p-form connections]; Belokurov & Shavgulidze a1109 [quantum field theories on loop space, local limit]; Afriat a1311 [on the reality of loops].
@ Statistical ensembles of loops: Troyer et al PRL(08) [quantum loop gas]; Nahum et al PRL(13) [in a 3D or higher-dimensional lattice, loop length distriution].
@ Gravity: Venkatesh a1212, a1305 [space and dynamics of gravity from loop algebras]; Nelson & Picken ATMP(14)-a1309 [intersecting loops on a 2D torus]; > s.a. loop quantum gravity.
@ Loop transform: Abbati et al LMP(01)mp [abelian group]; > used in quantum gauge theory; canonical quantum gravity.
> Other applications: see gauge theories [loop-based variables]; QCD; quantum field theory; string theory.