There is something strange with the definition as written. It says that the relations are in vector space, while they define a quotient Lie algebra. This does not parse. Namely, these are two very different statements, relations in a vector space close to a vector subspace. Relations in a Lie algebra have to be completed to a Lie ideal to make sense of a quotient. It is rather rare that the linear span of finitely many relations is the same as the Lie ideal generated by those relations.

So what is the intended statement ?

]]>the infinitesimal braid relations as stated weren’t quite right in arbitrary “dimension”.

For the moment I have fixed this (here) by declaing the dimension to be 2 and adding a warning that there is a more general definition.

]]>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)

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.)

]]>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+313Fred 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)

added illustration in terms of horizontal chord diagrams

]]>starting something – not done yet

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