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Video AKT-091006

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Lie algebraic weight systems, gl_N.

# 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.
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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0:00:02 [edit] Where we are in the diagram: usual knots, low algebra (Lie-algebra)

Goal: Given a finite-dimensional, metrized Lie algebra \mathcal{G} and a finite-dimensional representation R, we will construct a functional W_{\mathcal{G},R}: A(O) \rightarrow k (the base field).

0:03:35 [edit] Example: Let the Lie algebra be gl_n(k) (with [ \cdot, \cdot] being the commutator and \left\langle A, B \right\rangle= tr(AB))

Let the representation R: \rho: \mathcal{G} \rightarrow End(R), \rho([A, B]) = \rho(A) \circ \rho(B) - \rho(B) \circ \rho(A)

0:05:34 [edit] Recall the definition of \mathcal{A}: chord diagrams on a circle with trivalent points mod AS, IHX, STU relations.
0:08:29 [edit] Overview: we are now going to summarize the Lie-algebraic information in terms of tensors.

Warning: there will be many indices.

(Note: This can also be done abstractly, i.e. without choosing a basis, but then we can`t calculate anything.)

0:11:24 [edit] Choose a basis (X_a)^{\dim{\mathcal{G}}}_{a=1} for \mathcal{G} and a basis (e_{\alpha})^{\dim(R)}_{\alpha=1} for R.


[X_a, X_b] = f_{a,b}^c X_c, where f_{ab}^c \in \mathbb{Q} are the structure constants
\left\langle X_a, X_b\right\rangle = t_{ab}
Symmetric: t_{ab}=t_{ba}
Non-degenerate: (t_{ab}) has an inverse, (t^{ab}), with t_{ab} \cdot t^{bc} = \delta_{ac}
0:17:34 [edit] Expressing invariance and anti-symmetry of [\cdot, \cdot], <\cdot, \cdot> in terms of basis.
0:20:44 [edit] Expression representation, Jacobi identity and "r \cdot f = r_1 r_2 - r_2 r_1"
0:27:28 [edit] Define the map W_{\mathcal{G}, R}: \mathcal{A}(O) \rightarrow \mathbb{Q}

(we do it on a sample chord diagram)

0:33:00 [edit] Claim: W_{\mathcal{G}, R} as above is well defined.
0:36:44 [edit] Check that W_{\mathcal{G}, R} satisfies AS, IHX and STU relations. (They are all encoded in the Lie-algebra structure)
0:39:36 [edit] Computing W_{\mathcal{G}, R} with \mathcal{G} = gl_n (as described earlier in the class).