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# AKT-140303 Video

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Notes on AKT-140303:    [edit, refresh]

Lie algebraic weight systems.

# Week of... Notes and Links
: Course introduction, knots and Reidemeister moves, knot colourings.
Tricolourability without Diagrams
: The Gauss linking number combinatorially and as an integral.
: The Schroedinger equation and path integrals.
Friday Introduction (the quantum pendulum)
2 Jan 13 Homework Assignment 1.
: The Kauffman bracket and the Jones polynomial.
: Euler-Lagrange problems, Gaussian integration, volumes of spheres.
3 Jan 20 Homework Assignment 2.
: The definition of finite-type and some examples.
: The self-linking number and framings.
: Integrating a polynomial times a Gaussian.
Class Photo.
4 Jan 27 Homework Assignment 3.
: Chord diagrams and weight systems.
: Swaddling maps and framings, general configuration space integrals.
: Some analysis of $d^{-1}$.
5 Feb 3 Homework Assignment 4.
: 4T, the Fundamental Theorem and universal finite type invariants.
The Fulton-MacPherson Compactification (PDF).
: The Fulton-MacPherson Compactification, Part I.
: More on pushforwards, $d^{-1}$, and $d^\ast$.
6 Feb 10 Homework Assignment 5.
: The bracket-rise theorem and the invariance principle.
: The Fulton-MacPherson Compactification, Part II.
: Gauge fixing, the beginning of Feynman diagrams.
7 Feb 24 : A review of Lie algebras.
: Graph cohomology and $\Omega_{dR}^\ast(\Gamma)$.
: More on Feynman diagrams, beginning of gauge theory.
From Gaussian Integration to Feynman Diagrams (PDF).
8 Mar 3 Homework Assignment 6 (PDF)
: Lie algebraic weight systems.
: Graph cohomology and the construction of $Z_0$.
Graph Cohomology and Configuration Space Integrals (PDF)
: Gauge invariance, Chern-Simons, holonomies.
Mar 9 is the last day to drop this class.
9 Mar 10 Homework Assignment 7 (PDF)
: The $gl(N)$ weight system.
: The universal property, hidden faces.
: Insolubility of the quintic, naive expectations for CS perturbation theory.
10 Mar 17 Homework Assignment 8 (PDF)
: $W_{\mathfrak g}\colon{\mathcal A}(\uparrow)\to{\mathcal U}({\mathfrak g})$ and PBW.
: The anomaly.
Gaussian Integration, Determinants, Feynman Diagrams (PDF).
11 Mar 24 Homework Assignment 9 (PDF)
: ${\mathcal A}$ is a bi-algebra.
: Understanding and fixing the anomaly.
Friday: class cancelled.
12 Mar 31 Monday, Wednesday: class cancelled.
: A Monday class: back to expansions.
E Apr 7 : A Friday class on what we mostly didn't have time to do.

Dror's Notebook
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0:04:21 [add] Reminders: the structure constants.
0:16:01 [add] The construction of $W_{{\mathfrak g},R}$.
0:17:02 [add] Informal `universallity' of the construction.
0:21:42 [add] Well-definededness of $W_{{\mathfrak g},R}$.
0:21:43  Here we check for the cyclic symmetry of the structure coefficient $f$. First, we want to show that $f$ is total antisymmetric. For the first antisymmetry relation (i.e., the one with first two indices exchanged), we use antisymmetry relation of Lie algebra.

$f_{abc}=\langle\left[X_a,X_b\right],X_c\rangle=\langle -\left[X_b,X_a\right],X_c\rangle=-\langle\left[X_b,X_a\right],X_c\rangle=-f_{bac}.$

For the second antisymmetry relation (i.e., the one with last two indices exchanged), we use the fact the metric is invariant. A metric $\langle\cdot,\cdot\rangle:\mathfrak{g}\times\mathfrak{g}\rightarrow \mathbb{R}$ for any finite-dimensional Lie algebra $\mathfrak{g}$ is invariant

$\langle\left[z,x\right],y\rangle=-\langle x,\left[z,y\right]\rangle, \forall x,y,z\in\mathfrak{g}.$

Then, with the fact that the metric is symmetric, we write this relation in terms of structural coeffiecnt $f$.

$f_{abc}=\langle\left[X_a,X_b\right],X_c\rangle=-\langle X_b,\left[X_a,X_c\right]\rangle=-\langle \left[X_a,X_c\right],X_b\rangle=-f_{acb}.$

Thus, this shows the antisymmetry relations of $f$. Then, $f_{abc}=-f_{bac}=-\left(-f_{bca}\right)=f_{bca}$. Similarly, it shows $f_{bca}=f_{cab}$. Thus, the cyclic relation follows.

0:32:44 [add] Well-definededness of $W_{{\mathfrak g},R}$ (2).
0:35:03  Value of $W_{\mathfrak{g},R}$ on $IHX$.

Computation for $I$

\begin{align} W_{\mathfrak{g},R}(I) & = f_{ecd}f_{abe^{\prime}}t^{ee^{\prime}} \\ & = \langle[X_e, X_c], X_d \rangle \langle[X_a, X_b], X_e^{\prime} \rangle t^{ee^{\prime}} = f_{ec}^st_{sd} f_{ab}^kt_{ke^{\prime}}t^{ee^{\prime}}\\ & = f_{ec}^st_{sd} f_{ab}^k \delta_{k}^{e} \\ & = f_{ec}^sf_{ab}^et_{sd} \end{align}

Here $s$ is a dummy variable and could be replace by $e^{\prime}$

0:48:19 [add] $W_{{\mathfrak g},R}$ satisfies IHX, AS, STU.
0:48:28  After the end of the recording we had a further discussion of the fact that "not all invariant tensors in ${\mathfrak g}^{\otimes n}$ arise from a $W_{\mathfrak g}$-like construction". See also Haviv's note, where a symmetric invariant tensor in ${\mathfrak g}^{\otimes 3}$ is described for ${\mathfrak g}=sl(n)$, $n\geq 3$. Yet it is known that every "diagrammatic" element of ${\mathfrak g}^{\otimes 3}$ is anti-symmetric.
0:52:02 [add] Not all invariant tensors arise this way.