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nLab > Latest Changes: all horizontal weight systems are partitioned Lie algebra weight systems

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeDec 2nd 2019

    to record the result of

    v1, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeDec 2nd 2019
    • (edited Dec 2nd 2019)

    But I find this confusing, maybe somebody can help me:

    By Bar-Natan 96 all weight systems on horizontal chord diagrams are Lie.

    By Vogel 11 not all weight systems on round chord diagrams are Lie (nor even super-Lie).

    But by Kohno 02 (leading up to Theorem 4.2) the round weight systems are a subspace of the horizontal weight system.

    !?

    There must be some subtle fine print here, and now am worried that I am missing it.

    I guess the point is that Kohno 02 really talks about framed round weight systems (not imposing the 1T relation), while Vogel 11 speaks about unframed weight systems (imposing the 1T relation).

    (?)

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeDec 3rd 2019

    Oh, maybe I see: The weight systems in Bar-Natan 96 are not strictly Lie weight systems: The right map in Fact 7 is, but not its composition with Delta, giving the left map.

    So the theorem here really says something like “weighted” or “partitioned” Lie weight systems span all horizontal weight systems.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeDec 3rd 2019

    edited the entry to bring out this point about the weight systems here really being Lie algebra weight systems composed with a “partitioning” operation. Renamed the entry accordingly.

    diff, v2, current

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeDec 3rd 2019

    Hm, I suppose I need to include the choice of permutation in the “partitioning” data. I was thinking this is redundant, but now I see that it’s not. But need to go offline now.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeDec 3rd 2019

    have edited accordingly. The final statement now reads as follows:

    Span((||𝔰𝔩(N)Mod /∼⏟Liemodules)×(||⊕ℕℕ⏟tuplesofnumbers)×ℕ(⊔n∈ℕSym(n)⏟permutations)) ⟶epimorphismtr (−)∘w (−)∘Δ (−)(−) 𝒲 pbhorizontalweightsystemsa (C,k=(k 1,⋯,k n),σ) ↦ (Dhorizontalchorddiagrama↦ =tr σ⏞σ-trace∘W C ⊗k 1,⋯,C ⊗k n(D)⏟RTinvariant =tr σ∘w C⏟End-valuedLiealgebraweightsystem∘Δ k⏞partitioning(D)). \array{ Span \Big( \big( \underset{ \mathclap{ \color{blue} {Lie\,modules} } }{ \underbrace{ \mathclap{\phantom{\vert \atop \vert}} \mathfrak{sl}(N) Mod_{/\sim} } } \big) \; \times \; \big( \underset{ \mathclap{ \color{blue} tuples\;of\;numbers } }{ \underbrace{ \mathclap{ \phantom{\vert \atop \vert } } \underset{\mathbb{N}}{\oplus} \mathbb{N} } } \big) \; \underset{\mathbb{N}}{\times} \; \big( \underset{ \color{blue} permutations }{ \underbrace{ \underset{n \in \mathbb{N}}{\sqcup} Sym(n) } } \big) \Big) & \underoverset{\color{blue}epimorphism}{ \;\;\; tr_{(-)} \circ w_{(-)} \circ \Delta^{(-)} (-) \;\;\; }{\longrightarrow} & \overset{ \mathclap{ {\color{blue} horizontal\;weight\;systems} \atop {\phantom{a}} } }{ \mathcal{W}_{pb} } \\ ( C, \;\; k = (k_1, \cdots, k_n), \;\; \sigma ) &\mapsto& \left( \;\;\;\;\; \array{ \overset{ \mathclap{ \color{blue} { {horizontal} \atop { {chord} \atop {diagram} } } \atop {\phantom{a}} } }{ D } \mapsto & \phantom{=\;} \overset{ \mathclap{ \color{blue} \sigma\text{-}trace } }{ \overbrace{ tr_\sigma } } \circ \underset{ \mathclap{ {\color{blue}RT\;invariant} } }{ \underbrace{ W_{{}_{C^{\otimes k_1}, \cdots , C^{\otimes k_n} }}(D) } } \;\;\;\;\;\;\;\;\; \\ & = tr_\sigma \circ \underset{ \mathclap{ {\color{blue} End\text{-}valued\;Lie\;algebra\;weight\;system} } }{ \underbrace{ w_C } } \circ \overset{ \mathclap{ {\color{blue} partitioning} } }{ \overbrace{ \Delta^k } } (D) } \right) } \,.

    diff, v3, current

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeDec 3rd 2019
    • (edited Dec 3rd 2019)

    added illustrating example of the Δ\Delta-operation (here)

    diff, v4, current

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeDec 3rd 2019

    added also a more generic example

    diff, v5, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeDec 4th 2019
    • (edited Dec 4th 2019)

    added illustration of the final result (here)

    diff, v7, current

1 to 9 of 9

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