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nLab > Latest Changes: infinitesimal braid relation

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  1.  
    • 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”

    ℒ n(D)≔F({t ij} i≠j∈{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 pb≔Span(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(ℒ 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

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeSep 9th 2023
    • (edited Sep 9th 2023)

    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.

    diff, v8, current

    • CommentRowNumber7.
    • CommentAuthorzskoda
    • CommentTimeSep 11th 2023
    • (edited Sep 11th 2023)

    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 ?

1 to 7 of 7

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