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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeNov 26th 2019

    starting something – not done yet

    v1, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeNov 26th 2019

    added illustration in terms of horizontal chord diagrams

    v1, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeDec 17th 2019

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

    diff, v4, current

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

    made explicit the “infinitesimal braid Lie algebra”

    n(D)F({t ij} ij{1,,n})/(R0,R1,R2). \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 ijt_{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

    (𝒜 pbSpan(𝒟 n pb)/(2T,4T),) \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:

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

    (now this prop.)

    diff, v4, current

    • 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)

    diff, v5, current

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