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Video AKT-090929

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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
1 Sep 7 About This Class
dbnvp 090910-1: 3-colourings, Reidemeister's theorem, invariance, the Kauffman bracket.
dbnvp 090910-2: R23 invariance of the bracket, R1, the writhe, the Jones polynomial, programming the Jones polynomial.
Tricolourability
2 Sep 14 dbnvp 090915: More on Jones, some pathologies and more on Reidemeister, our overall agenda.
dbnvp 090917-1: The definition of finite type, weight systems, Jones is a finite type series.
dbnvp 090917-2: The skein relation for Jones; HOMFLY-PT and Conway; the weight system of Jones.
3 Sep 21 dbnvp 090922: FI, 4T, HOMFLY and FI and 4T, statement of the Fundamental Theorem, framed knots.
dbnvp 090924-1: Some dimensions of {\mathcal A}_n, {\mathcal A} is a commutative algebra, {\mathcal A}(\bigcirc)\equiv{\mathcal A}(\uparrow).
Class Photo
dbnvp 090924-2: {\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
dbnvp 090929: The Milnor-Moore theorem, primitives, the map {\mathcal A}^r\to{\mathcal A}.
dbnvp 091001-1: Jacobi diagrams, AS, IHX, STU, and the equivalence of all that with 4T.
dbnvp 091001-2: The very basics on Lie algebras.
5 Oct 5 dbnvp 091006: Lie algebraic weight systems, gl_N.
dbnvp 091008-1: More on gl_N, Lie algebras and the four colour theorem.
dbnvp 091008-2: The "abstract tenssor" approach to weight systems, {\mathcal U}({\mathfrak g}) and PBW, the map {\mathcal T}_{\mathfrak g}.
6 Oct 12 dbnvp 091013: Algebraic properties of {\mathcal U}({\mathfrak g}) vs. algebraic properties of {\mathcal A}.
Thursday's class canceled.
7 Oct 19 dbnvp 091020: Universal finite type invariants, filtered and graded spaces, expansions.
Homework Assignment 2
The Stonehenge Story
dbnvp 091022-1: The Stonehenge Story to IHX and STU.
dbnvp 091022-2: The Stonhenge Story: anomalies, framings, relation with physics.
8 Oct 26 dbnvp 091027: Knotted trivalent graphs and their chord diagrams.
dbnvp 091029-1: Zsuzsi Dancso on the Kontsevich Integral (1).
dbnvp 091029-2: Zsuzsi Dancso on the Kontsevich Integral (2).
9 Nov 2 dbnvp 091103: The details of {\mathcal A}^{TG}.
dbnvp 091105-1: Three basic problems: genus, unknotting numbers, ribbon knots.
dbnvp 091105-2: The three basic problems and algebraic knot theory.
10 Nov 9 dbnvp 091110: Tangles and planar algebras, shielding and the generators of KTG.
Homework Assignment 3
No Thursday class.
11 Nov 16 Local Khovanov Homology
dbnvp 091119-1: Local Khovanov homology, I.
dbnvp 091119-2: Local Khovanov homology, II.
12 Nov 23 dbnvp 091124: Emulation of one structure inside another, deriving the pentagon.
dbnvp 091126-1: Peter Lee on braided monoidal categories, I.
dbnvp 091126-2: Peter Lee on braided monoidal categories, II.
13 Nov 30 dbnvp 091201: The relations in KTG.
dbnvp 091203-1: The Existence of the Exponential Function.
dbnvp 091203-2: The Final Exam, Dror's failures.
F Dec 7 The Final Exam on Thu Dec 10, 9-11, Bahen 6183.
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0:00:18 [edit] See Kolmogorov's Solution of Hilbert's 13th Problem.
0:02:38 [edit] See Embedding 3-Manifolds in 4-Space.
0:03:56 [edit] 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 [edit] 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 [edit] 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 [edit] 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 [edit] examples of primitive elements in \mathcal{A}
0:14:58 [edit] The universal enveloping algebra of an commutative graded connected bi-algebra is the bi-algebra of polynomials on its primitives.
0:18:15 [edit] 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 [edit] See a similar table at 06-1350/Class_Notes_for_Thursday_October_12.
0:26:40 [edit] See HW1.

(tiny piece of Milnor-Moore theorem)

0:37:23 [edit] 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 [edit] 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 [edit] Comment by Peter about the map P.
0:47:42 [edit] Further description and hint