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Video AKT-091001-2

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The very basics on Lie algebras.

# 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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0:00:25 [edit] Aim: Make the relationship between \mathcal{A} and Lie algebras formal and move into low algebra.
0:01:11 [edit] Before that, some combinatorics: \mathcal{A}^t is a bialgebra with

  1. Product: concatenation.
  2. Coproduct: sum over all ways of distributing the connected components, when the skeleton is ignored, between the two sides of the tensor product.
0:05:19 [edit] Verify that the co-product satisfies the STU relation.
0:09:23 [edit] Corollary: Diagrams with only one connected component (after removing the outer circle) are primitive. In fact one can show those diagrams span the space of primitives.
0:13:25 [edit] Theorem:

1) Given a metrized, finite-dimensional Lie algebra \mathcal{G} and a finite-dimensional representation R of \mathcal{G}, \exists (an interesting) linear functional W_{\mathcal{G},R}: \mathcal{A} \rightarrow \mathbb{Q} (the ground field on which \mathcal{G} is defined).

0:15:43 [edit] 2) Given \mathcal{G} (metrized finite dim Lie algebra) \exists\ {\mathcal T}_\mathcal{G}: \mathcal{A}(|) \rightarrow U(\mathcal{G}) where U(\mathcal{G}) is the universal enveloping algebra of \mathcal{G}.
0:17:42 [edit] Definition of Lie-algebra
0:20:46 [edit] Examples:

1.\mathcal{G}=\mathbb{Q}^n with [ \cdot , \cdot ] = \overline{0}

2. \mathcal{G} = gl_n with [A, B]=AB-BA

3. \mathcal{G} = set of all anti-symmetric matrices in gl_n with the bracket as defined in gl_n

4. FL_2(x,y)= set of all trees with leaves labeled with symbols x and y mod AS and STU with [\cdot, \cdot] being the operation that connects the roots of the trees

5. Given two Lie-algrbras, the direct sum of them is a Lie-algrbra.

0:28:08 [edit] A metric of a Lie-algebra is a nondegenerate symmetric bilinear form < \cdot, \cdot >: \mathcal{G} \otimes \mathcal{G} \rightarrow \mathbb{Q} s.t. <[z,x], y> + <x, [z,y]> = 0
0:31:29 [edit] Examples of metrics (the interesting one being on the lie-algebra gl_n, define <A,B> = tr(AB))
0:37:29 [edit] Definition: A representation of Lie-algebra \mathcal{G} is a Lie-algebra morphism R: \mathcal{G} \rightarrow End(V_R) (endomorphisms of some vector space V_R)
0:42:00 [edit] If R_1, R_2 are representations of \mathcal{G}, then so is R_1 \oplus R_2 and R_1 \otimes R_2 (here R_1, R_2 denotes the vector space s.t. End(R) is the target space of R)