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Video AKT-091013

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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
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.
Register of Good Deeds / To Do List
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0:00:06 [edit] Announcements: 1. Dror is sick today 2.next class should be FUN
0:01:20 [edit] 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 [edit] 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 [edit] 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 [edit] \mathcal{T}_\mathcal{G} is not onto and maps \mathcal{A}(|) into the invariant part of U(\mathcal{G}).
0:27:10 [edit] Generalizing \mathcal{T}_\mathcal{G} to diagrams with multiple baselines
0:31:23 [edit] When n>1, \mathcal{A}(|_n) is a non-commutative algebra.
0:35:09 [edit] When n>1, invariant part of U(\mathcal{G})^{\otimes n} is strictly larger than Z(U(\mathcal{G})^{\otimes n})
0:36:40 [edit] Fact 2: \mathcal{A}(|) as a co-algebra (Define \Delta: \mathcal{A}(|) \rightarrow \mathcal{A}(|_2))
0:41:59 [edit] 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 [edit] Fact 4 (PBW): Diagrams/STU is isomorphic to diagrams with baseline erased / AS, IHX