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

# Video AKT-091008-2

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The "abstract tenssor" approach to weight systems, ${\mathcal U}({\mathfrak g})$ and PBW, the map ${\mathcal T}_{\mathfrak g}$.

# 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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0:00:03  Aside: relation between $\mathcal{A}(\phi)$ and 3-manifold invariants.
0:01:32  Redo the previous construction without fixing basis. (i.e. we can do everything in a more abstract setting by looking at abstract tensors in various tensor products of $\mathcal{G}$ and $\mathcal{G}^*$)
0:09:54  Computation of value on diagrams using abstract tensors. (breaking up the diagram and connect back with contractions)
0:13:58  Participation sheet
0:15:17  Weight systems without a representation:

Tensor algebra (of lie algebra $\mathcal{G}$): $T(\mathcal{G})=\{$ words with letters in $\mathcal{G} \}/$linearity in the letters

Universal enveloping algebra (of $\mathcal{G}$): $U(\mathcal{G})=T(\mathcal{G})/<[x,y]=xy-yx>$

0:21:16  PBW theorem: Choose an ordered basis $x_1, \cdots, x_n$ of $\mathcal{G}$, then $U(\mathcal{G})$ is the vector space with basis being all monotone non-decreasing words in $x_1, \cdots, x_n$
0:26:47  sketch of the proof of PBW theorem:

Given any word, go through each letter and change the ordering of the letters by applying the relation $[x,y]=xy-yx$ this process will yield an expression of the word using only non-decreasing words in the basis elements.

0:30:35  Problem: it's not clear that the procedure is well-defined (i.e. does the final expression depend on the order of sorting?)

solution: write the terms explicitly and apply the Jacobi identity

0:36:32  Claim: Given Lie-algebra $\mathcal{G}$ and metric $t$, we have $\mathcal{T}_\mathcal{G}: \mathcal{A}(|) \rightarrow U(\mathcal{G})$ s.t. given any representation $R$, $\exists \ \ tr_R: U(\mathcal{G}) \rightarrow \mathbb{Q},\ \ tr_R \circ \mathcal{T}_\mathcal{G}=W_{\mathcal{G},R}$.
0:38:11  Defining $\mathcal{T}_\mathcal{G}$
0:41:06  Checking that $\mathcal{T}_\mathcal{G}$ is well-defined under AS, IHX and STU relation:

AS and IHX is internal (does not touch the base line)hence satisfied by construction, STU becomes the relation $[x,y]=xy-yx$

0:42:36  Defining map $tr_R: U(\mathcal{G}) \rightarrow \mathbb{Q}$
0:44:43  preview of next time: Looking back to combinatorics and topology.