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Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
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)
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.)
added pointer to what seems to be the original reference:
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 ?
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