Drorbn - User contributions [en]

Notes for AKT-091020/0:42:10

2009-10-22T02:56:27Z

Conan777:

Claim: Expansions always exist.

Notes for AKT-091020/0:38:54

2009-10-22T02:53:37Z

Conan777:

Definition of expansion

Notes for AKT-091020/0:37:29

2009-10-22T02:52:42Z

Conan777:

gr <math>\circ</math> fil <math>\cong</math>(naturally equivalent) Id,

but fil <math>\circ</math> gr is not naturally equivalent to identity.

Notes for AKT-091020/0:37:29

2009-10-22T02:51:50Z

Conan777:

gr <math>\circ</math> fil <math>\cong</math> (naturally equivalent) Id, but fil <math>\circ</math> gr is not.

Notes for AKT-091020/0:37:29

2009-10-22T02:49:38Z

Conan777:

gr <math>\circ</math> fil <math>\cong</math> Id

Notes for AKT-091020/0:34:31

2009-10-22T02:48:01Z

Conan777:

Define funtor '''Fil''': g-vect <math>\rightarrow</math> F-vect

and funtor '''gr''': F-vect <math>\rightarrow</math> g-vect

Notes for AKT-091020/0:34:31

2009-10-22T02:47:43Z

Conan777:

Define funtor '''Fil''': g-vect <math>\rightarrow</math> F-vect
and funtoe '''gr''': F-vect <math>\rightarrow</math> g-vect

Notes for AKT-091020/0:34:31

2009-10-22T02:46:59Z

Conan777:

Define funtor '''Fil''': g-vect <math>\rightarrow</math> F-vect

Notes for AKT-091020/0:34:31

2009-10-22T02:46:47Z

Conan777:

Define funtor '''Fil''': g-vect \rightarrow F-vect

Notes for AKT-091020/0:29:55

2009-10-22T02:45:20Z

Conan777:

F-vect: category of filtered vector spaces

Notes for AKT-091020/0:32:25

2009-10-22T02:44:49Z

Conan777:

g-vect: category of graded vector spaces

Notes for AKT-091020/0:29:55

2009-10-22T02:41:14Z

Conan777:

F-vut: category of filtered vector spaces

Notes for AKT-091020/0:22:01

2009-10-22T02:37:38Z

Conan777:

Let <math>Z(K)=\Sigma_n \Sigma_{i=1}^{\dim(\mathcal{A}_n)} D_{n,i}V_{n,i}(K)</math>

Notes for AKT-091020/0:19:13

2009-10-22T02:34:26Z

Conan777:

Fundamental theorem <math>\Rightarrow \ \exists</math> UFTI <math>Z</math>

For all <math>n</math>, pick basis <math>W_{n, i}</math> of <math>\mathcal{A}_n^*</math>, by fundamental theorem, there are type <math>n</math> invariants <math>V_{n, i}</math> where <math>V_{n, i}^{(n)}=W_{n,i}</math>

Let <math>D_{n,i}</math> be the dual basis of <math>W_{n, i}</math>

Notes for AKT-091020/0:15:25

2009-10-22T02:26:17Z

Conan777:

Karnen's question: Does <math>V</math> also vanish on knots of type <math> < n</math>? (No)

Notes for AKT-091020/0:11:41

2009-10-22T02:18:38Z

Conan777:

<math>V</math> is of type <math>n</math> and <math>W</math> is the weight system of <math>V</math>

Notes for AKT-091020/0:10:00

2009-10-22T02:17:24Z

Conan777:

Proof: 1) <math>\ \exists</math> UFTI <math>Z \ \ \Rightarrow \ </math> fundamental theorem

Given <math>W \in \mathcal{A}_n^*</math>, set <math>V = W \circ Z</math>

Notes for AKT-091020/0:10:00

2009-10-22T02:01:01Z

Conan777:

Proof: 1) <math>\ \exists</math> UFTI <math>Z \ \ \Rightarrow \ </math> fundamental theorem

Notes for AKT-091020/0:08:39

2009-10-22T01:58:31Z

Conan777:

Claim: the fundamental theorem holds iff universal finite type exists.

Notes for AKT-091020/0:05:39

2009-10-22T01:57:42Z

Conan777:

Def: a '''universal finite type invariant''' is an invariant <math>Z: \mathcal{K} \rightarrow \mathcal{A}</math> s.t. for <math>n</math>-singular <math>K</math>, <math>Z(K) = D_K +</math> diagrams of higher degree. (Note that a universal finite type invariant is '''not''' an finite type invariant)

Notes for AKT-091020/0:02:23

2009-10-22T01:52:25Z

Conan777:

Reminder:

Fundamental theorem: Every weight system integrates.

Notes for AKT-091020/0:01:05

2009-10-22T01:48:38Z

Conan777:

Warning: the class is going to be extremely boring

Notes for AKT-091013/0:44:56

2009-10-22T01:44:16Z

Conan777:

Fact 4 (PBW): Diagrams/STU is isomorphic to diagrams with baseline erased / AS, IHX

Notes for AKT-091013/0:44:56

2009-10-22T01:42:22Z

Conan777:

Fact 4 (PBW): Diagrams/STU corresponding to diagrams with baseline erased

Notes for AKT-091013/0:41:59

2009-10-22T01:37:04Z

Conan777:

Fact 3: Given <math>\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}</math> under the canonical isomorphism.

Notes for AKT-091013/0:36:40

2009-10-22T01:28:47Z

Conan777:

Fact 2: <math>\mathcal{A}(|)</math> as a co-algebra (Define <math>\Delta: \mathcal{A}(|) \rightarrow \mathcal{A}(|_2)</math>)

Notes for AKT-091013/0:35:09

2009-10-22T01:24:44Z

Conan777:

When <math>n>1</math>, invariant part of <math>U(\mathcal{G})^{\otimes n}</math> is strictly larger than <math>Z(U(\mathcal{G})^{\otimes n})</math>

Notes for AKT-091013/0:31:23

2009-10-22T01:18:43Z

Conan777:

When <math>n>1</math>, <math>\mathcal{A}(|_n)</math> is a non-commutative algebra.

Notes for AKT-091013/0:27:10

2009-10-22T01:13:14Z

Conan777:

Generalizing <math>\mathcal{T}_\mathcal{G}</math> to diagrams with multiple baselines

Notes for AKT-091013/0:17:29

2009-10-22T00:50:31Z

Conan777:

<math>\mathcal{T}_\mathcal{G}</math> is not onto and maps <math>\mathcal{A}(|)</math> into the invariant part of <math>U(\mathcal{G})</math>.

Notes for AKT-091013/0:13:01

2009-10-22T00:47:09Z

Conan777:

Relate back to <math>\mathcal{A}(|)</math>:

Fact 1: <math>\mathcal{A}(|)</math> is an algebra with multiplication being composing two diagrams

Note that this algebra is commutative

Notes for AKT-091013/0:03:17

2009-10-22T00:31:02Z

Conan777:

Facts about <math>U(\mathcal{G})</math>:

1. <math>U(\mathcal{G})</math> is an (non-communicative) algebra

2. <math>U(\mathcal{G})</math> is a co-algebra

3. <math>U(\mathcal{G}_1 \oplus \mathcal{G}_2) \cong U(\mathcal{G}_1) \otimes U(\mathcal{G}_2)</math>

4. <math>U(\mathcal{G}) \cong S(\mathcal{G})</math> as co-algebras and <math>\mathcal{G}</math> modules

Notes for AKT-091013/0:01:20

2009-10-21T00:48:46Z

Conan777:

Locating where we are: towards the end of the first pass on low algebra, will soon declare the goal for the high algebra part.

Notes for AKT-091013/0:00:06

2009-10-21T00:46:06Z

Conan777:

Announcements: 1. Dror is sick today 2.next class should be FUN

Notes for AKT-091008-2/0:44:43

2009-10-21T00:43:13Z

Conan777:

preview of next time: Looking back to combinatorics and topology.

Notes for AKT-091008-2/0:42:36

2009-10-21T00:40:38Z

Conan777:

Defining map <math>tr_R: U(\mathcal{G}) \rightarrow \mathbb{Q}</math>

Notes for AKT-091008-2/0:41:06

2009-10-21T00:38:48Z

Conan777:

Checking that <math>\mathcal{T}_\mathcal{G}</math> 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 <math>[x,y]=xy-yx</math>

Notes for AKT-091008-2/0:38:11

2009-10-21T00:35:54Z

Conan777:

Defining <math>\mathcal{T}_\mathcal{G}</math>

Notes for AKT-091008-2/0:36:32

2009-10-21T00:29:45Z

Conan777:

Claim: Given Lie-algebra <math>\mathcal{G}</math> and metric <math>t</math>, we have <math>\mathcal{T}_\mathcal{G}: \mathcal{A}(|) \rightarrow U(\mathcal{G})</math> s.t. given any representation <math>R</math>, <math>\exists \ \ tr_R: U(\mathcal{G}) \rightarrow \mathbb{Q},\ \ tr_R \circ \mathcal{T}_\mathcal{G}=W_{\mathcal{G},R}</math>.

Notes for AKT-091008-2/0:30:35

2009-10-21T00:21:05Z

Conan777:

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

Notes for AKT-091008-2/0:30:35

2009-10-21T00:17:59Z

Conan777:

Problem: it's not clear that the procedure is well-defined (i.e. does the final expression depend on the order of sorting?)

Notes for AKT-091008-2/0:26:47

2009-10-21T00:15:16Z

Conan777:

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 <math>[x,y]=xy-yx</math> this process will yield an expression of the word using only non-decreasing words in the basis elements.

Notes for AKT-091008-2/0:21:16

2009-10-21T00:09:10Z

Conan777:

PBW theorem: Choose an ordered basis <math>x_1, \cdots, x_n</math> of <math>\mathcal{G}</math>, then <math>U(\mathcal{G})</math> is the vector space with basis being all monotone non-decreasing words in <math>x_1, \cdots, x_n</math>

Notes for AKT-091008-2/0:15:17

2009-10-21T00:02:02Z

Conan777:

Weight systems without a representation:

Tensor algebra (of lie algebra <math>\mathcal{G}</math>): <math>T(\mathcal{G})=\{</math> words with letters in <math>\mathcal{G} \}/</math>linearity in the letters

Universal enveloping algebra (of <math>\mathcal{G}</math>): <math>U(\mathcal{G})=T(\mathcal{G})/<[x,y]=xy-yx>
</math>

Notes for AKT-091008-2/0:15:17

2009-10-20T23:53:08Z

Conan777:

Weight systems without a representation:

Definition of the universal enveloping algebra of a given lie algebra

Notes for AKT-091008-2/0:13:58

2009-10-20T23:50:25Z

Conan777:

Participation sheet

Notes for AKT-091008-2/0:09:54

2009-10-20T23:48:26Z

Conan777:

Computation of value on diagrams using abstract tensors. (breaking up the diagram and connect back with contractions)

Notes for AKT-091008-2/0:01:32

2009-10-20T23:43:41Z

Conan777:

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 <math>\mathcal{G}</math> and <math>\mathcal{G}^*</math>)

Notes for AKT-091008-2/0:01:32

2009-10-20T23:41:32Z

Conan777:

Redo the previous construction without fixing basis. (i.e. we can do everything in a more abstract setting by looking at various tensor products of <math>\mathcal{G}</math> and <math>\mathcal{G}^*</math>)

Notes for AKT-091008-2/0:00:03

2009-10-20T23:31:34Z

Conan777:

Aside: relation between <math>\mathcal{A}(\phi)</math> and 3-manifold invariants.