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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeNov 26th 2019
   • PermaLink
   Author: Urs
   Format: MarkdownItexstarting something -- not done yet <a href="https://ncatlab.org/nlab/revision/infinitesimal+braid+relation/1">v1</a>, <a href="https://ncatlab.org/nlab/show/infinitesimal+braid+relation">current</a>

   starting something – not done yet

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeNov 26th 2019
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded illustration in terms of horizontal chord diagrams <a href="https://ncatlab.org/nlab/revision/infinitesimal+braid+relation/1">v1</a>, <a href="https://ncatlab.org/nlab/show/infinitesimal+braid+relation">current</a>

   added illustration in terms of horizontal chord diagrams

   v1, current

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeDec 17th 2019
   • PermaLink
   Author: Urs
   Format: MarkdownItexfixed a sign in the definition, and added a bunch of references: * [[Dror Bar-Natan]], Fact 3 in: _Vassiliev and Quantum Invariants of Braids_, Geom. Topol. Monogr. 4 (2002) 143-160 ([arxiv:q-alg/9607001](https://arxiv.org/abs/q-alg/9607001)) * [[Toshitake Kohno]], _Linear representations of braid groups and classical Yang-Baxter equations_, Cont. Math. 78 (1988), 339-363. * [[Edward Fadell]], [[Sufian Husseini]], Theorem 2.2 in: _Geometry and topology of configuration spaces_, Springer Monographs in Mathematics (2001), [MR2002k:55038](http://www.ams.org/mathscinet-getitem?mr=2002k:55038), xvi+313 * [[Fred Cohen]], [[Samuel Gitler]], Section 3 in: _Loop spaces of configuration spaces, braid-like groups, and knots_, In: Jaume Aguadé, Carles Broto, [[Carles Casacuberta]] (eds.) _Cohomological Methods in Homotopy Theory_. Progress in Mathematics, vol 196. Birkhäuser, Basel 2001 ([doi:10.1007/978-3-0348-8312-2_7](https://doi.org/10.1007/978-3-0348-8312-2_7)) * [[Fred Cohen]], [[Samuel Gitler]], p. 2 of: _On loop spaces of configuration spaces_, Trans. Amer. Math. Soc. __354__ (2002), no. 5, 1705&#8211;1748, ([jstor:2693715](https://www.jstor.org/stable/2693715), [MR2002m:55020](http://www.ams.org/mathscinet-getitem?mr=1881013)) <a href="https://ncatlab.org/nlab/revision/diff/infinitesimal+braid+relation/4">diff</a>, <a href="https://ncatlab.org/nlab/revision/infinitesimal+braid+relation/4">v4</a>, <a href="https://ncatlab.org/nlab/show/infinitesimal+braid+relation">current</a>

   fixed a sign in the definition, and added a bunch of references:

   • Dror Bar-Natan, Fact 3 in: Vassiliev and Quantum Invariants of Braids, Geom. Topol. Monogr. 4 (2002) 143-160 (arxiv:q-alg/9607001)

   • Toshitake Kohno, Linear representations of braid groups and classical Yang-Baxter equations, Cont. Math. 78 (1988), 339-363.

   • Edward Fadell, Sufian Husseini, Theorem 2.2 in: Geometry and topology of configuration spaces, Springer Monographs in Mathematics (2001), MR2002k:55038, xvi+313

   • Fred Cohen, Samuel Gitler, Section 3 in: Loop spaces of configuration spaces, braid-like groups, and knots, In: Jaume Aguadé, Carles Broto, Carles Casacuberta (eds.) Cohomological Methods in Homotopy Theory. Progress in Mathematics, vol 196. Birkhäuser, Basel 2001 (doi:10.1007/978-3-0348-8312-2_7)

   • Fred Cohen, Samuel Gitler, p. 2 of: On loop spaces of configuration spaces, Trans. Amer. Math. Soc. 354 (2002), no. 5, 1705–1748, (jstor:2693715, MR2002m:55020)

   diff, v4, current

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeDec 17th 2019
   • (edited Dec 17th 2019)
   • PermaLink
   Author: Urs
   Format: MarkdownItexmade explicit the "infinitesimal braid Lie algebra" $$ \mathcal{L}_n(D) \;\coloneqq\; F(\{t_{i j}\}_{i\neq j \in \{1,\cdots, n\}}) /(R0, R1, R2) \,. $$ being the quotient of the free Lie algebra on the generators $t_{i j}$ modulo the infinitesimal braid relations (now [this Def.](https://ncatlab.org/nlab/show/infinitesimal+braid+relation#InfinitesimalBraidLieAlgebra)) Then I made more explicit the algebra of horizontal chord diagrams modulo 2T- and 4T-relations $$ \Big( \mathcal{A}^{pb} \;\coloneqq\; Span \big( \mathcal{D}_n^{pb} \big)/(2T, 4T) , \circ \Big) $$ and its equivalence to the universal enveloping algebra of the infinitesimal braid Lie algebra: $$ \big(\mathcal{A}_n^{pb}, \circ\big) \;\simeq\; \mathcal{U}(\mathcal{L}_n(D)) \,. $$ (now [this prop.](https://ncatlab.org/nlab/show/infinitesimal+braid+relation#UniversalEnvelopingOfInfinitesimalBraidsIsHorizontalChords)) <a href="https://ncatlab.org/nlab/revision/diff/infinitesimal+braid+relation/4">diff</a>, <a href="https://ncatlab.org/nlab/revision/infinitesimal+braid+relation/4">v4</a>, <a href="https://ncatlab.org/nlab/show/infinitesimal+braid+relation">current</a>

   made explicit the “infinitesimal braid Lie algebra”

   $$\mathcal{L}_n(D) \;\coloneqq\; F(\{t_{i j}\}_{i\neq j \in \{1,\cdots, n\}}) /(R0, R1, R2) \,.$$

   being the quotient of the free Lie algebra on the generators $t_{i j}$ modulo the infinitesimal braid relations (now this Def.)

   Then I made more explicit the algebra of horizontal chord diagrams modulo 2T- and 4T-relations

   $$\Big( \mathcal{A}^{pb} \;\coloneqq\; Span \big( \mathcal{D}_n^{pb} \big)/(2T, 4T) , \circ \Big)$$

   and its equivalence to the universal enveloping algebra of the infinitesimal braid Lie algebra:

   $$\big(\mathcal{A}_n^{pb}, \circ\big) \;\simeq\; \mathcal{U}(\mathcal{L}_n(D)) \,.$$

   (now this prop.)

   diff, v4, current

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeDec 18th 2019
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded pointer to what seems to be the original reference: * [[Toshitake Kohno]], (1.1.4) in: _Monodromy representations of braid groups and Yang-Baxter equations_, Annales de l'Institut Fourier, Volume 37 (1987) no. 4, p. 139-160 ([doi:10.5802/aif.1114](https://doi.org/10.5802/aif.1114)) <a href="https://ncatlab.org/nlab/revision/diff/infinitesimal+braid+relation/5">diff</a>, <a href="https://ncatlab.org/nlab/revision/infinitesimal+braid+relation/5">v5</a>, <a href="https://ncatlab.org/nlab/show/infinitesimal+braid+relation">current</a>

   added pointer to what seems to be the original reference:

   • Toshitake Kohno, (1.1.4) in: Monodromy representations of braid groups and Yang-Baxter equations, Annales de l’Institut Fourier, Volume 37 (1987) no. 4, p. 139-160 (doi:10.5802/aif.1114)

   diff, v5, current