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all horizontal weight systems are partitioned Lie algebra weight systems

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Knot theory

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∞-Lie theory (higher geometry)

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Contents

Idea

All weight systems on horizontal chord diagrams may be realized as linear combinations of Lie algebra weight systems applied not necessarily to the given horizontal chord diagram itself, but to the result of regarding each of its strands as resolved by some finite number of strands.

We state this precisely as Prop. below (due to Bar-Natan 96). First we introduce all the definitions that enter the statement:

Ingredients

Given any ground field 𝔽\mathbb{F} (or in fact just any commutative ground ring)

1) write 𝒜 pb\mathcal{A}^{pb} for the linear span of horizontal chord diagrams modulo the 2T relations and the 4T relations

regarded as a graded vector space, graded by the number of chords,

and write

(1)𝒲 pb(𝒜 pb) * \mathcal{W}_{pb} \coloneqq (\mathcal{A}^{pb})^\ast

for its degreewise dual vector space: the space of horizontal weight systems;

2) write

𝒜 pbp n𝒜 n pbi n𝒜 pb \mathcal{A}^{pb} \overset{ \;\;\; p_n \;\;\; }{\longrightarrow} \mathcal{A}^{pb}_n \overset{ \;\;\; i_n \;\;\; }{\hookrightarrow} \mathcal{A}^{pb}

for projection onto and inclusion of the linear subspace spanned by horizontal chord diagrams with nn strands;

3) write

Δ ()End(𝒜 pb) \underset{\mathbb{N}}{\oplus} \mathbb{N} \overset{ \;\;\; \Delta^{(-)} \;\;\; }{\longrightarrow} End( \mathcal{A}^{pb} \big)

for the operation that reads in a finite tuple k(k 1,,k n)k \coloneqq (k_1, \cdots, k_n) of natural numbers, with sum |k|sumik i\left\vert k\right\vert \coloneqq \underset{i}{sum} k_i, and produces the linear map

(2)𝒜 pbp n𝒜 n pbΔ k𝒜 |k| pbi |k|𝒜 pb \mathcal{A}^{pb} \overset{ \; p_n \; }{\to} \mathcal{A}^{pb}_{n} \overset{ \;\;\; \Delta^k \;\;\; }{\longrightarrow} \mathcal{A}^{pb}_{\left\vert k \right \vert} \overset{ \; i_{\left\vert k \right\vert} \; }{\hookrightarrow} \mathcal{A}^{pb}

which takes a horizontal chord diagram with nn strands to the linear combination of chord diagrams obtained by replacing its ii-th strand by k ik_i strands for all ii and then summing over all ways of re-attaching chords, with any vertex previously on some strand ii now to be put on one of the k ik_i strands (Bar-Natan 96, Def. 2.2).

For example:


Moreover, for 𝔤\mathfrak{g} a metric Lie algebra

1) write

𝔤Mod / \mathfrak{g}Mod_{/\sim}

for its set of isomorphism classes of finite dimensional Lie algebra representations (Lie modules)

2) write

(3)𝔤Mod /w ()nHom 𝔽(𝒜 n pb,End(C n)) \mathfrak{g}Mod_{/\sim} \overset{ w_{(-)} }{\longrightarrow} \underset{n \in \mathbb{N}}{\oplus} Hom_{\mathbb{F}} \big( \mathcal{A}^{pb}_n , End(C^{\otimes n}) \big)

for the function that sends a Lie module CC over 𝔤\mathfrak{g} to the corresponding endomorphism ring-valued Lie algebra weight system w Cw_C on horizontal chord diagrams.

Finally, for

  1. C𝔤ModC \in \mathfrak{g} Mod a Lie algebra representation of 𝔤\mathfrak{g},

  2. nn \in \mathbb{N} a natural number,

  3. σSym(n)\sigma \in Sym(n) a permutation of nn elements

write

(4)tr σ:End(C n)𝔽 tr_\sigma \;\colon\; End \big( C^{\otimes n} \big) \longrightarrow \mathbb{F}

for the composite operation of

  1. composing an endomorphism on the nn-fold tensor power of CC by the braiding according to the permutation σ\sigma;

  2. forming the trace of the resulting endomorphism of C nC^{\otimes n}.

Then the composition of

  1. the partitioning function (2);

  2. the assignment (3) of Lie algebra weight systems;

  3. the permuted trace operation (4)

yields a function from triples consisting of a Lie module, a tuple of natural numbers and a permutation to horizontal weight systems:

(5)(𝔤Mod /)×()×(nSym(n)) tr ()w ()Δ 𝒲 pb (C,k=(k 1,,k n),σ) (Dtr σw CΔ k(D)) \array{ \big( \mathfrak{g}Mod_{/\sim} \big) \;\times\; \big( \underset{\mathbb{N}}{\oplus} \mathbb{N} \big) \; \underset{ \mathbb{N} }{\times} \; \big( \underset{n \in \mathbb{N}}{\sqcup} Sym(n) \big) & \overset{ \;\; tr_{(-)} \circ w_{(-)} \circ \Delta \;\; }{ \longrightarrow } & \mathcal{W}_{pb} \\ (C, \; k = (k_1, \cdots, k_n), \; \sigma) &\mapsto& \left( D \;\mapsto\; tr_\sigma \circ w_C \circ \Delta^k (D) \right) }

Finally, write also

(6)Span(𝔤Mod /××nSym(n))tr ()w ()Δ ()()𝒲 pb Span \big( \mathfrak{g}Mod_{/\sim} \;\times\; \underset{\mathbb{N}}{\oplus} \mathbb{N} \;\underset{\mathbb{N}}{\times}\; \underset{n \in \mathbb{N}}{\mathbb{N}} Sym(n) \big) \overset{ tr_{(-)} \circ w_{(-)} \circ \Delta^{(-)} (-) }{ \longrightarrow } \mathcal{W}_{pb}

for the linear extension of this function (5) to the linear span of its domain set.


Statement

Proposition

(all horizontal weight systems are partitioned Lie algebra weight systems)

For N2N \geq 2 consider the special linear Lie algebra 𝔰𝔩(N)\mathfrak{sl}(N), canonically regarded as a metric Lie algebra via its Killing form.

Then the space 𝒲 pb(𝒜 pb) *\mathcal{W}_{pb} \coloneqq (\mathcal{A}^{pb})^\ast (1) of weight systems on horizontal chord diagrams is spanned by partitioned 𝔰𝔩(N)\mathfrak{sl}(N)-Lie algebra weight systems, in that the linear extension (6) of the function (5) assigning 𝔰𝔩(N)\mathfrak{sl}(N)-Lie algebra weight systems composed with partitioning (2) is an epimorphism:

Span((||𝔰𝔩(N)Mod /Liemodules)×(||tuplesofnumbers)×(nSym(n)permutations)) epimorphismtr ()w ()Δ ()() 𝒲 pbhorizontalweightsystemsa (C,k=(k 1,,k n),σ) (Dhorizontalchorddiagrama =tr σσ-traceW C k 1,,C k n(D)RTinvariant =tr σw CEnd-valuedLiealgebraweightsystemΔ kpartitioning(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) } \,.

This is the statement of Bar-Natan 96, Corollary 2.6.

For example:

References

The theorem and its proof is due to:

Last revised on December 4, 2019 at 02:52:47. See the history of this page for a list of all contributions to it.