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- Loop_group abstract "In mathematics, a loop group is a group of loops in a topological group G with multiplication defined pointwise. Specifically, let LG denote the space of continuous mapsequipped with the compact-open topology. An element of is called a loop in G. Pointwise multiplication of such loops gives the structure of a topological group. The space is called the free loop group on . A loop group is any subgroup of the free loop group .An important example of a loop group is the group of based loops on . It is defined to be the kernel of the evaluation map ,and hence is a closed normal subgroup of . (Here, is the map that sends a loop to its value at .) Note that we may embed into as the subgroup of constant loops. Consequently, we arrive at a split exact sequence .The space splits as a semi-direct product, .We may also think of as the loop space on . From this point of view, is an H-space with respect to concatenation of loops. On the face of it, this seems to provide with two very different product maps. However, it can be shown that concatenation and pointwise multiplication are homotopic. Thus, in terms of the homotopy theory of , these maps are interchangeable.Loop groups were used to explain the phenomenon of Bäcklund transforms in soliton equations by Chuu-Lian Terng and Karen Uhlenbeck.".
- Loop_group wikiPageExternalLink books?id=MbFBXyuxLKgC.
- Loop_group wikiPageID "1390288".
- Loop_group wikiPageRevisionID "596116255".
- Loop_group hasPhotoCollection Loop_group.
- Loop_group subject Category:Solitons.
- Loop_group subject Category:Topological_groups.
- Loop_group type Abstraction100002137.
- Loop_group type Event100029378.
- Loop_group type Group100031264.
- Loop_group type Happening107283608.
- Loop_group type Movement107309781.
- Loop_group type PsychologicalFeature100023100.
- Loop_group type Soliton107346344.
- Loop_group type Solitons.
- Loop_group type TopologicalGroups.
- Loop_group type TravelingWave107347051.
- Loop_group type Wave107345593.
- Loop_group type YagoPermanentlyLocatedEntity.
- Loop_group comment "In mathematics, a loop group is a group of loops in a topological group G with multiplication defined pointwise. Specifically, let LG denote the space of continuous mapsequipped with the compact-open topology. An element of is called a loop in G. Pointwise multiplication of such loops gives the structure of a topological group. The space is called the free loop group on .".
- Loop_group label "Loop group".
- Loop_group sameAs m.04yrb_.
- Loop_group sameAs Q6675827.
- Loop_group sameAs Q6675827.
- Loop_group sameAs Loop_group.
- Loop_group wasDerivedFrom Loop_group?oldid=596116255.
- Loop_group isPrimaryTopicOf Loop_group.