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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeNov 25th 2019
   • PermaLink
   Author: Urs
   Format: MarkdownItexstarting something on weight systems (in the sense of knot theory), not done yet <a href="https://ncatlab.org/nlab/revision/weight+system/1">v1</a>, <a href="https://ncatlab.org/nlab/show/weight+system">current</a>

   starting something on weight systems (in the sense of knot theory), not done yet

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeNov 25th 2019
   • PermaLink
   Author: Urs
   Format: MarkdownItexCombining the two propositions 1. [[weight systems are associated graded of Vassiliev invariants]], 1. [[weight systems are cohomology of loop space of configuration space]] we get [this](https://ncatlab.org/nlab/files/VassilievInvInCohOfLoopSpacesOfConfigSpaces.jpg) situation. Hope I got this right... <a href="https://ncatlab.org/nlab/revision/diff/weight+system/3">diff</a>, <a href="https://ncatlab.org/nlab/revision/weight+system/3">v3</a>, <a href="https://ncatlab.org/nlab/show/weight+system">current</a>

   Combining the two propositions

   1. weight systems are associated graded of Vassiliev invariants,

   2. weight systems are cohomology of loop space of configuration space

   we get this situation.

   Hope I got this right…

   diff, v3, current

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeNov 27th 2019
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded a pointer to [[Lie algebra weight systems]]: *** A large class of [[weight systems]] arises from reading a ([[horizontal chord diagram|horizontal]]) [[chord diagram]] as a [[string diagram]] in the evident way, and then labelling it by the structure morphisms of a [[Lie algebra object]] equipped with a [[Lie algebra representation]] [[internalization|internal to]] a suitable [[tensor category]]. This does yield weight systems because the required relations translate exactly to the structural equations satisfied by [[Lie modules]] ([[Jacobi identity]] and Lie action property). The weight systems arising this way are called _[[Lie algebra weight systems]]_. See there for more <a href="https://ncatlab.org/nlab/revision/diff/weight+system/6">diff</a>, <a href="https://ncatlab.org/nlab/revision/weight+system/6">v6</a>, <a href="https://ncatlab.org/nlab/show/weight+system">current</a>

   added a pointer to Lie algebra weight systems:

   ---

   A large class of weight systems arises from reading a (horizontal) chord diagram as a string diagram in the evident way, and then labelling it by the structure morphisms of a Lie algebra object equipped with a Lie algebra representation internal to a suitable tensor category. This does yield weight systems because the required relations translate exactly to the structural equations satisfied by Lie modules (Jacobi identity and Lie action property).

   The weight systems arising this way are called Lie algebra weight systems. See there for more

   diff, v6, current

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeDec 8th 2019
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded pointer to * [[Vladimir Hinich]], [[Arkady Vaintrob]], _Cyclic operads and algebra of chord diagrams_, Sel. math., New ser. (2002) 8: 237 ([arXiv:math/0005197](https://arxiv.org/abs/math/0005197)) <a href="https://ncatlab.org/nlab/revision/diff/weight+system/12">diff</a>, <a href="https://ncatlab.org/nlab/revision/weight+system/12">v12</a>, <a href="https://ncatlab.org/nlab/show/weight+system">current</a>

   added pointer to

   • Vladimir Hinich, Arkady Vaintrob, Cyclic operads and algebra of chord diagrams, Sel. math., New ser. (2002) 8: 237 (arXiv:math/0005197)

   diff, v12, current

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeDec 28th 2019
   • (edited Dec 28th 2019)
   • PermaLink
   Author: Urs
   Format: MarkdownItexre-organized subsections: split off "Constructions of weight systems" from "Properties"; added stub section on [[stringy weight systems]] <a href="https://ncatlab.org/nlab/revision/diff/weight+system/15">diff</a>, <a href="https://ncatlab.org/nlab/revision/weight+system/15">v15</a>, <a href="https://ncatlab.org/nlab/show/weight+system">current</a>

   re-organized subsections: split off “Constructions of weight systems” from “Properties”; added stub section on stringy weight systems

   diff, v15, current