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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeNov 26th 2019

starting something – not done yet

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeNov 26th 2019

added illustration in terms of horizontal chord diagrams

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeDec 17th 2019

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

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeDec 17th 2019
• (edited Dec 17th 2019)

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.)

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeDec 18th 2019

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)
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