Not signed in (Sign In)

Start a new discussion

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry beauty bundle bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics comma complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched etcs fibration foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie lie-theory limits linear linear-algebra locale localization logic manifolds mathematics measure measure-theory modal modal-logic model model-category-theory monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory string string-theory subobject superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • 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

Add your comments
  • Please log in or leave your comment as a "guest post". If commenting as a "guest", please include your name in the message as a courtesy. Note: only certain categories allow guest posts.
  • To produce a hyperlink to an nLab entry, simply put double square brackets around its name, e.g. [[category]]. To use (La)TeX mathematics in your post, make sure Markdown+Itex is selected below and put your mathematics between dollar signs as usual. Only a subset of the usual TeX math commands are accepted: see here for a list.

  • (Help)