Cheat Sheet OneCo  http://drorbn.net/AcademicPensieve/2015-05/ initiated 14/4/15; modified 10/6/15, 4:43am; continues 2015-05

Deriving Gassner. L^{2Dw} is b_{i} ⟨a_{ij}⟩ modulo locality, [a_{ij}, a_{ik}] = 0, [a_{ik}, a_{jk}] = −[a_{ij}, a_{jk}] = b_{j}a_{ik} − b_{i}a_{jk}, and (mod ⟨a_{ii}⟩) [a_{ij}, a_{ji}] = b_{i}a_{ji} − b_{j}a_{ij}. Acts on V = b_{i} ⟨x_{i} = a_{i∞} ⟩ by [a_{ij}, x_{i}] = 0, [a_{ij}, x_{j}] = b_{i}x_{j} − b_{j}x_{i}. Hence e^{ada_{ij}}x_{i} = x_{i}, e^{ada_{ij}}x_{j} = e^{b_{i}}x_{j} + \frac{b_{j}}{b_{i}}(1 − e^{b_{i}})x_{i}. Renaming y_{i} = x_{i}/b_{i}, t_{i} = e^{b_{i}}, get

[e^{ada_{ij}}]_{y_{i},y_{j}} = (\begin{matrix}1 & 1 − t_{i} \\ 0 & t_{i} \end{matrix}).

The L^{2Dw} Adjoint representation. e^{ada_{ij}} acts by

a_{kl} ↦ a_{kl},  a_{ik} ↦ a_{ik},  a_{kj} ↦ e^{−b_{i}}a_{kj} + \frac{b_{k}}{b_{i}}(1 − e^{−b_{i}})a_{ij},

a_{ki} ↦ a_{ki} + (1 − e^{−b_{i}})a_{kj} + b_{k} \frac{e^{−b_{i}} − 1}{b_{i}} a_{ij},

a_{jk} ↦ e^{b_{i}}a_{jk} + \frac{b_{j}}{b_{i}}(1 − e^{b_{i}})a_{ik}, a_{ji} ↦ e^{b_{i}}a_{ji} + \frac{b_{j}}{b_{i}}(1 − e^{b_{i}})a_{ij}.

Adjoint Gassner. Renaming α_{ij} = a_{ij}/b_{i} and t_{i} = e^{b_{i}}, get

α_{kj} ↦ t_{i}^{−1}α_{kj} + (1 − t_{i}^{−1})α_{ij},

α_{ki} ↦ a_{ki} + (1 − t_{i}^{−1})α_{kj} + (t_{i}^{−1} − 1)α_{ij}

α_{jk} ↦ t_{i}α_{jk} + (1 − t_{i})α_{ik}, α_{ji} ↦ t_{i}α_{ji} + (1 − t_{i})α_{ij}.

Implementation/verification: pensieve://2015-04/nb/ZeroCo.pdf . Interpretation: π_{T}-Artin?

2Dv. b: bracket trace; c: cobracket trace; ⟨b, c⟩ = δ ∈ {0, 1} ; degb_{i} = degc_{j} = dega_{ij} = degδ = 1. A^{2Dv} is δ FA(b_{i}, c_{j}, a_{ij}) (so L^{v} = {f +f^{ij}a_{ij}}) modulo locality, tt. (note γ_{j{kl}}= 0) [a_{jk}, a_{jl}] = c_{l}a_{jk} − c_{k}a_{jl} γ_{jkl} , hh. [a_{jk}, a_{ik}] = b_{i}a_{jk} − b_{j}a_{ik}, ht. [a_{jk}, a_{kl}] = b_{j}a_{kl} − b_{k}a_{jl} − γ_{jkl} , . [a_{ij}, a_{ji}] = ?, ab, ac. [a_{ij}, b_{i}] = −[a_{ij}, b_{j}] = −[a_{ij}, c_{i}] = [a_{ij}, c_{j}]

= δa_{ij} − b_{i}c_{j} γ_{ij},

bc. [b_{i}, c_{j}] = 0.

So a_{ij}f = f^{δ}a_{ij} − \frac{b_{i}c_{j}}{δ}(f^{δ} − f), [a_{ij}, f] = (f^{δ} − f) (a_{ij} − \frac{b_{i}c_{j}}{δ}),

with f^{δ} f⫽\binom{b_{i}→b_{i}+δ b_{j}→b_{j}−δ}{c_{i}→c_{i}−δ c_{j}→c_{j}+δ}.

Repeating indices convention. In a_{ii}, γ_{ii}, γ_{iij}, γ_{ijj}, and γ_{iii}, heads are above tails. The Ascending Algebra A_{+}^{2Dv}. Same but with only a_{ij}, i < j.

The OneCo Sub-Quotient is ⟨a_{ij}⟩ modulo δ^{2} = δc_{i} = c_{j}c_{k} = 0, so L^{1co} is (coefficient functions non-central, in b_{i} )

{f^{ij}a_{ij} + f^{ijk}γ_{ijk} + f^{ijkl}γ_{ij}a_{kl}} / (b_{i}γ_{ijk} = γ_{ij}a_{ik} − γ_{ik}a_{ij})_{(is there a residual 4T?)} .

Representations. ada_{−,−}:

[figure: ⟨β⟩; ⟨γ_{ij}⟩; ⟨a_{ij}⟩; ⟨γ_{ijk}⟩; ⟨γ_{ij}a_{kl}⟩]

• ⟨γ_{ij}⟩ is …

To do. • Perhaps I should find a way to highlight the fact that v is a perturbation of w. • Position FiC. • Position the 2D Lie bialgebras. • Is there a meaningful a_{ij} → a_{ij}/b_{i} (etc) renormalization?

• Add: diagrammatic interpretations of b_{i}, c_{j}, γ_{ij}, γ_{ijk}.

Recycling. Models. • In [x, y] = δx, xf (y) = f (y + δ)x. If δ^{2} = 0, [x, f (y)] = δf′(y)x.

• In [x, y] = δx + z^{2}, xf (y) = f (y + δ)x + \frac{z^{2}}{δ}(f (y + δ) − f (y)). If δ^{2} = 0, [x, f (y)] = δf′(y)x + z^{2}f′(y).

• If S_{n} ∑ _{k=0}^{n−1} A^{k}CB^{n−1−k} then AS_{n} − S_{n}B = A^{n}C − CB^{n} so S_{n} = (L_{A}− R_{B})^{−1}(A^{n}C − CB^{n}).

• If ψ(x) = ∑ _{n≥0} a_{n}x^{n} then ∑ _{n≥0} a_{n}∑ _{k=0}^{n−1} b^{n}(−b)^{n−1−k} = (ψ(b) − ψ(−b))/2b . The primitivity condition. ker(f + f^{ij}a_{ij} ↦ δf + f^{ij}b_{i}c_{j}) . (Ignoring multiple arrows). State Diagrams. ada_{ij} yields green: roots. pink: wrong.

[figure: f c_{i}; b̂_{i}; −; f c_{j}; +; −b̂_{i}; δf a_{ij}; f; b̂_{i}(∂_{j}−∂_{i}); ∂_{i}−∂_{j}; a_{jk}; b̂_{i}; +; −; −b̂_{j}; c_{j}a_{ik}; −b̂_{i}; +; c_{k}a_{ij}; a_{ik}; +; −; δa_{ij}a_{ik}]

[figure: a_{kj}; −b̂_{i}; b̂_{k}; a_{ij}]

so with ϕ_{0} ϕ(0), ϕ_{1} ϕ′_{0}, and ϕ_{↓}(x) (ϕ(x) − ϕ_{0})/x, ϕ(ada_{ij}) is

f c_{i} ↦ ϕ_{0}f c_{i} + (b_{i}ϕ_{↓↓}(−b_{i}) − ϕ_{1})δf a_{ij} + b_{i}ϕ_{↓}(−b_{i})f c_{j}
f c_{j} ↦ ϕ(−b_{i})f c_{j} + ϕ_{↓}(−b_{i})δf a_{ij}
f ↦ ϕ_{0}f + b_{i}ϕ_{↓}(−b_{i})(∂_{j}f − ∂_{i}f)c_{j}

+ (b_{i}ϕ_{↓↓}(−b_{i}) − ϕ_{1})(∂_{j}f − ∂_{i}f)δa_{ij}

δa_{··}↦ as in Adjoint Gassner
a_{ik} ↦ ϕ_{0}a_{ik} + ϕ_{1}c_{k}a_{ij} − ϕ_{↓}(−b_{i})c_{j}a_{ik} − ϕ_{↓↓}(−b_{i})δa_{ij}a_{ik}
a_{jk} ↦ ϕ(b_{i})a_{jk} − (ϕ_{↓}(b_{i}) + b_{j}ϕ_{↓↓}(b_{i}))c_{k}a_{ij} − b_{j}ϕ_{↓}(b_{i})a_{ik}

+ \frac{ϕ(b_{i}) − ϕ(−b_{i}) + b_{j}(ϕ_{↓}(b_{i}) − ϕ_{↓}(−b_{i}))}{2b_{i}} c_{j}a_{ik}

+ \frac{ϕ_{↓}(b_{i}) − ϕ_{↓}(−b_{i}) + b_{j}(ϕ_{↓↓}(b_{i}) − ϕ_{↓↓}(−b_{i}))}{2b_{i}} δa_{ij}a_{ik}

a_{kj} ↦
a_{ij} ↦ a_{ij}
Then [a_{ij}, f] = (∂_{i}f − ∂_{j}f)γ_{ij} and

γb. [γ_{ij}, b_{l}] = 0 and [γ_{ijk}, b_{l}] = 0 incl. l ∈ {i, j, k}, ttγ. [a_{jk}, γ_{jl}] = 0, hhγ. [a_{jk}, γ_{ik}] = −b_{j}γ_{ik}, thγ. [a_{jk}, γ_{ij}] = b_{j}γ_{ik}, htγ. [a_{jk}, γ_{kl}] = b_{j}γ_{kl} − b_{k}γ_{jl}, ttγ_{3}. [a_{jk}, γ_{jlm}] = 0, thγ_{3}. [a_{jk}, γ_{ijl}] = b_{j}γ_{ikl} + γ_{il}a_{jk}, htγ_{3}. [a_{jk}, γ_{klm}] = b_{k}γ_{jkl} + b_{j}γ_{klm}, hhγ_{3}. [a_{jk}, γ_{nik}] = −b_{j}γ_{nik}+ γ_{ni}a_{jk},

[a_{jk}, γ_{jkl}] = −γ_{jk}a_{jl},

[a_{jk}, γ_{ijk}] = −b_{j}γ_{ijk} + γ_{ij}a_{jk} + γ_{ik}a_{jk}.