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

# Video AKT-091013

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Videography by Karene Chu

Notes on AKT-091013:    [edit, refresh]

Algebraic properties of ${\mathcal U}({\mathfrak g})$ vs. algebraic properties of ${\mathcal A}$.

# 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.
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0:00:06  Announcements: 1. Dror is sick today 2.next class should be FUN
0:01:20  Locating where we are: towards the end of the first pass on low algebra, will soon declare the goal for the high algebra part.
0:03:17  Facts about $U(\mathcal{G})$:

1. $U(\mathcal{G})$ is an (non-communicative) algebra

2. $U(\mathcal{G})$ is a co-algebra

3. $U(\mathcal{G}_1 \oplus \mathcal{G}_2) \cong U(\mathcal{G}_1) \otimes U(\mathcal{G}_2)$

4. $U(\mathcal{G}) \cong S(\mathcal{G})$ as co-algebras and $\mathcal{G}$ modules

0:13:01  Relate back to $\mathcal{A}(|)$:

Fact 1: $\mathcal{A}(|)$ is an algebra with multiplication being composing two diagrams

Note that this algebra is commutative

0:17:29  $\mathcal{T}_\mathcal{G}$ is not onto and maps $\mathcal{A}(|)$ into the invariant part of $U(\mathcal{G})$.
0:27:10  Generalizing $\mathcal{T}_\mathcal{G}$ to diagrams with multiple baselines
0:31:23  When $n>1$, $\mathcal{A}(|_n)$ is a non-commutative algebra.
0:35:09  When $n>1$, invariant part of $U(\mathcal{G})^{\otimes n}$ is strictly larger than $Z(U(\mathcal{G})^{\otimes n})$
0:36:40  Fact 2: $\mathcal{A}(|)$ as a co-algebra (Define $\Delta: \mathcal{A}(|) \rightarrow \mathcal{A}(|_2)$)
0:41:59  Fact 3: Given $\mathcal{G}_1, \ \mathcal{G}_2, \ \ \mathcal{T}_{\mathcal{G}_1} \otimes \mathcal{T}_{\mathcal{G}_2} \circ \Box = \mathcal{T}_{\mathcal{G}_1 \oplus \mathcal{G}_2}$ under the canonical isomorphism.
0:44:56  Fact 4 (PBW): Diagrams/STU is isomorphic to diagrams with baseline erased / AS, IHX