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

# Video AKT-090929

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Videography by Karene Chu

Notes on AKT-090929:    [edit, refresh]

The Milnor-Moore theorem, primitives, the map ${\mathcal A}^r\to{\mathcal A}$.

# 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.
Register of Good Deeds / To Do List

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0:03:56  From last time: $\mathcal{A}$ is the double dual of the space of knots, with product (connected sum of two chord diagrams) and co-product (Sum of all ways to split a chord diagram), it's a graded connected commutative co-commutative bi-algebra.
0:07:40  Graded: every element has a degree competitive w.r.t. the operations

Connected: A graded bi-algebra is connected if its degree zero sub-bi-algebra is $1$-dimensional

0:09:24  Theorem (Milnor-Moore): A graded, connected, co-commutative bialgebra is the universal enveloping algebra of its space of primitives:

$\mathcal{A}=U(\mathcal{P}(\mathcal{A}))$

(A proof is given here.)

0:10:43  Definition. Primitive: $\mathcal{P}(\mathcal{A})=\{a \in \mathcal{A}: \square a= a \otimes 1 + 1 \otimes a\}$, where $\square$ denotes the coproduct.
0:11:48  examples of primitive elements in $\mathcal{A}$
0:14:58  The universal enveloping algebra of an commutative graded connected bi-algebra is the bi-algebra of polynomials on its primitives.
0:18:15  Dimensions of $\mathcal{A}_m$, $\mathcal{A}_m^r$ and $\mathcal{P}_m$

$\dim(\mathcal{A}_m^r)=\dim(\mathcal{A}_m)-\dim(\mathcal{A}_{m-1})$

$\dim(\mathcal{P}_m)= \dim(\mathcal{A}_m) - |\{ \mbox{degree m products of lower degree primitive elements} \} |$

0:18:35  See a similar table at 06-1350/Class_Notes_for_Thursday_October_12.
0:26:40  See HW1.

(tiny piece of Milnor-Moore theorem)

0:37:23  Relation to quantum mechanics (Von Neumann's theorem): whenever the commutator of two operations equals the identity, one should be thought of as the derivative and the other as multiplication. Here the two operators are $\hat{\theta}$ - multiplication by $\theta$, and $\hat{W}^*_1$ - the adjoint of the dual of $\hat{\theta}$.
0:42:53  We can deduce that the projection $P:\mathcal{A} \rightarrow \mathcal{A}$ which takes $\theta$ to zero is given by:

$P=\sum^{\infty}_{n=0}\frac{(-\hat{\theta})^n}{n!}(\hat{W}^*_1)^n$
0:46:15  Comment by Peter about the map $P$.
0:47:42  Further description and hint