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Video AKT-091008-2

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Notes on AKT-091008-2:    [edit, refresh]

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
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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0:00:03 [edit] Aside: relation between \mathcal{A}(\phi) and 3-manifold invariants.
0:01:32 [edit] 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 [edit] Computation of value on diagrams using abstract tensors. (breaking up the diagram and connect back with contractions)
0:13:58 [edit] Participation sheet
0:15:17 [edit] 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 [edit] 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 [edit] 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 [edit] 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 [edit] 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 [edit] Defining \mathcal{T}_\mathcal{G}
0:41:06 [edit] 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 [edit] Defining map tr_R: U(\mathcal{G}) \rightarrow \mathbb{Q}
0:44:43 [edit] preview of next time: Looking back to combinatorics and topology.