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In its most general form a loop group is a group of continuous mappings from a manifold M to a topological group G.. More specifically, [1] let M = S 1, the circle in the complex plane, and let LG denote the space of continuous maps S 1 → G, i.e.
Similarly, a set of all smooth maps from S 1 to a Lie group G forms an infinite-dimensional Lie group (Lie group in the sense we can define functional derivatives over it) called the loop group. The Lie algebra of a loop group is the corresponding loop algebra.
Comments: When the inner mapping group Inn(Q) is finite and abelian, then Q is nilpotent (Niemenaa and Kepka). The first question is therefore open only in the infinite case. Call loop Q of Csörgõ type if it is nilpotent of class at least 3, and Inn(Q) is abelian. No loop of Csörgõ type of nilpotency class higher than 3 is known.
The edge-path group E(X, v) is defined to be the set of edge-equivalence classes of edge-loops at v, with product and inverse defined by concatenation and reversal of edge-loops. The edge-path group is naturally isomorphic to π 1 (|X |, v), the fundamental group of the geometric realisation |X | of X. [24]
If a loop is isotopic to a group, then it is isomorphic to that group and thus is itself a group. However, a quasigroup that is isotopic to a group need not be a group. For example, the quasigroup on R with multiplication given by ( x , y ) ↦ ( x + y )/2 is isotopic to the additive group ( R , +) , but is not itself a group as it has no ...
The loop representation is a quantum hamiltonian representation of gauge theories in terms of loops. The aim of the loop representation in the context of Yang–Mills theories is to avoid the redundancy introduced by Gauss gauge symmetries allowing to work directly in the space of physical states (Gauss gauge invariant states).
To be precise, the loop braid group on n loops is defined as the motion group of n disjoint circles embedded in a compact three-dimensional "box" diffeomorphic to the three-dimensional disk. A motion is a loop in the configuration space, which consists of all possible ways of embedding n circles into the 3-disk. This becomes a group in the same ...
Leeds loop, a traffic distribution route in the city centre of Leeds, United Kingdom; Loop Parkway, a spur of the Meadowbrook State Parkway on Long Island, New York, U.S. Massachusetts Route 213, a highway between I-93 and I-495 in Massachusetts, U.S. The Loop (Texarkana), a beltway around Texarkana, Texas and Arkansas, U.S.