© | Dror Bar-Natan: Classes: 2009-10: AKT:

# Video AKT-091006

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Videography by Karene Chu

Notes on AKT-091006:    [edit, refresh]

Lie algebraic weight systems, $gl_N$.

# Week of... Videos, Notes, and Links
: 3-colourings, Reidemeister's theorem, invariance, the Kauffman bracket.
: R23 invariance of the bracket, R1, the writhe, the Jones polynomial, programming the Jones polynomial.
Tricolourability
2 Sep 14 : More on Jones, some pathologies and more on Reidemeister, our overall agenda.
: The definition of finite type, weight systems, Jones is a finite type series.
: The skein relation for Jones; HOMFLY-PT and Conway; the weight system of Jones.
3 Sep 21 : FI, 4T, HOMFLY and FI and 4T, statement of the Fundamental Theorem, framed knots.
: Some dimensions of ${\mathcal A}_n$, ${\mathcal A}$ is a commutative algebra, ${\mathcal A}(\bigcirc)\equiv{\mathcal A}(\uparrow)$.
Class Photo
: ${\mathcal A}$ is a co-commutative algebra, the relation with products of invariants, ${\mathcal A}$ is a bi-algebra.
4 Sep 28 Homework Assignment 1
Homework Assignment 1 Solutions
: The Milnor-Moore theorem, primitives, the map ${\mathcal A}^r\to{\mathcal A}$.
: Jacobi diagrams, AS, IHX, STU, and the equivalence of all that with 4T.
: The very basics on Lie algebras.
5 Oct 5 : Lie algebraic weight systems, $gl_N$.
: More on $gl_N$, Lie algebras and the four colour theorem.
: The "abstract tenssor" approach to weight systems, ${\mathcal U}({\mathfrak g})$ and PBW, the map ${\mathcal T}_{\mathfrak g}$.
6 Oct 12 : Algebraic properties of ${\mathcal U}({\mathfrak g})$ vs. algebraic properties of ${\mathcal A}$.
Thursday's class canceled.
7 Oct 19 : Universal finite type invariants, filtered and graded spaces, expansions.
Homework Assignment 2
The Stonehenge Story
: The Stonehenge Story to IHX and STU.
: The Stonhenge Story: anomalies, framings, relation with physics.
8 Oct 26 : Knotted trivalent graphs and their chord diagrams.
: Zsuzsi Dancso on the Kontsevich Integral (1).
: Zsuzsi Dancso on the Kontsevich Integral (2).
9 Nov 2 : The details of ${\mathcal A}^{TG}$.
: Three basic problems: genus, unknotting numbers, ribbon knots.
: The three basic problems and algebraic knot theory.
10 Nov 9 : Tangles and planar algebras, shielding and the generators of KTG.
Homework Assignment 3
No Thursday class.
11 Nov 16 Local Khovanov Homology
: Local Khovanov homology, I.
: Local Khovanov homology, II.
12 Nov 23 : Emulation of one structure inside another, deriving the pentagon.
: Peter Lee on braided monoidal categories, I.
: Peter Lee on braided monoidal categories, II.
13 Nov 30 : The relations in KTG.
: The Existence of the Exponential Function.
: The Final Exam, Dror's failures.
F Dec 7 The Final Exam on Thu Dec 10, 9-11, Bahen 6183.
Register of Good Deeds / To Do List

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0:00:02  Where we are in the diagram: usual knots, low algebra (Lie-algebra)

Goal: Given a finite-dimensional, metrized Lie algebra $\mathcal{G}$ and a finite-dimensional representation $R$, we will construct a functional $W_{\mathcal{G},R}: A(O) \rightarrow k$ (the base field).

0:03:35  Example: Let the Lie algebra be $gl_n(k)$ (with $[ \cdot, \cdot]$ being the commutator and $\left\langle A, B \right\rangle= tr(AB)$)

Let the representation $R$: $\rho: \mathcal{G} \rightarrow End(R), \rho([A, B]) = \rho(A) \circ \rho(B) - \rho(B) \circ \rho(A)$

0:05:34  Recall the definition of $\mathcal{A}$: chord diagrams on a circle with trivalent points mod AS, IHX, STU relations.
0:08:29  Overview: we are now going to summarize the Lie-algebraic information in terms of tensors.

Warning: there will be many indices.

(Note: This can also be done abstractly, i.e. without choosing a basis, but then we can`t calculate anything.)

0:11:24  Choose a basis $(X_a)^{\dim{\mathcal{G}}}_{a=1}$ for $\mathcal{G}$ and a basis $(e_{\alpha})^{\dim(R)}_{\alpha=1}$ for $R$.

Notation:

$[X_a, X_b] = f_{a,b}^c X_c$, where $f_{ab}^c \in \mathbb{Q}$ are the structure constants
$\left\langle X_a, X_b\right\rangle = t_{ab}$
Symmetric: $t_{ab}=t_{ba}$
Non-degenerate: $(t_{ab})$ has an inverse, $(t^{ab})$, with $t_{ab} \cdot t^{bc} = \delta_{ac}$
0:17:34  Expressing invariance and anti-symmetry of $[\cdot, \cdot]$, $<\cdot, \cdot>$ in terms of basis.
0:20:44  Expression representation, Jacobi identity and "$r \cdot f = r_1 r_2 - r_2 r_1$"
0:27:28  Define the map $W_{\mathcal{G}, R}: \mathcal{A}(O) \rightarrow \mathbb{Q}$

(we do it on a sample chord diagram)

0:33:00  Claim: $W_{\mathcal{G}, R}$ as above is well defined.
0:36:44  Check that $W_{\mathcal{G}, R}$ satisfies AS, IHX and STU relations. (They are all encoded in the Lie-algebra structure)
0:39:36  Computing $W_{\mathcal{G}, R}$ with $\mathcal{G} = gl_n$ (as described earlier in the class).