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wClips-120314

Speaker: Karene Chu

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Section 3.6 - the relation with Lie algebras.

Announcements. small circle, UofT, LDT Blog (also here). Email Dror to join our mailing list!

Resources. How to use this site, Dror's notebook, blackboard shots.

The wClips

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Jan 11, 2012 : Introduction.
: Section 2.1 - v-Braids.
Jan 18, 2012 : An introduction to this web site.
: Section 2.2 - w-Braids by generators and relations and as flying rings.
: Section 2.2 - w-Braids - other drawing conventions, "wens".
Jan 25, 2012 : Section 2.2.3 - basis conjugating automorphisms of $F_n$.
: A very quick introduction to finite type invariants in the "u" case.
Feb 1, 2012 : Section 2.3 - finite type invariants of v- and w-braids, arrow diagrams, 6T, TC and 4T relations, expansions / universal finite type invariants.
Feb 8, 2012 : Review of u,v, and w braids and of Section 2.3.
Feb 15, 2012 : Section 2.5 - mostly compatibilities of $Z^w$, also injectivity and uniqueness of $Z^w$.
Feb 22, 2012 : Section 2.5.5, $\alpha:{\mathcal A}^u\to{\mathcal A}^v$, and Section 3.1 (partially), the definition of v- and w-knots.
Feb 29, 2012 : Sections 3.1-3.4: v-Knots and w-Knots: Definitions, framings, finite type invariants, dimensions, and the expansion in the w case.
Mar 7, 2012 : Section 3.5: Jacobi diagrams and the bracket-rise theorem.
Mar 14, 2012 : Section 3.6 - the relation with Lie algebras.
Mar 21, 2012 : Section 4 - Algebraic Structures.
Mar 28, 2012 Out-of-sequence not-on-tape we watched the video of Talks: GWU-1203.
Apr 4, 2012 : Section 3.7 - The Alexander Theorem (statement).
Apr 18, 2012 : Aside on the Euler trick, the differential of $\exp$, and the BCH formula.
Apr 25, 2012 : Section 3.8, a disorganized lecture towards the proof of the Alexander theorem.
May 2, 2012 : Section 4: Algebraic structures (review), circuit algebras, v- and w-tangles.
May 10, 2012 : Sections 5.1 and 5.2: tangles, their projectivization and its relationship with Alekseev-Torossian spaces.
May 23, 2012 : Section 5.2: Proof of the relationship with A-T spaces.
May 30, 2012 : Interpreting ${\mathcal A}^w(\uparrow_n)$ as a universal space of invariant tangential differential operators.
Group photo on January 11, 2012: DBN, ZD, Stephen Morgan, Lucy Zhang, Iva Halacheva, David Li-Bland, Sam Selmani, Oleg Chterental, Peter Lee.
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0:00:00  Blackboard shots by Zsuzsanna Dancso.

Sorry, for this wClip we've used a different video camera than the usual (the usual one was in ), and there are several bugs in this page as a result. --Drorbn 21:45, 20 March 2012 (EDT)

0:04:47  No need to forget about the algebra structure - this is an algebra isomorphism, and even a bialgebra isomorphism. --Drorbn 18:38, 19 March 2012 (EDT)
0:07:35 [add] ${\mathcal A}^w$ and ${\mathcal A}^{wt}$.
0:11:34  It is great to know the relationship with Lie bialgebras, as it is extremely relevant for the study of ${\mathcal A}^v$. Yet it is also good to know that $I{\mathfrak g}$ has a much simpler definition, that avoids some of the complexity. Namely, $I{\mathfrak g}$ is the semi-direct product ${\mathfrak g}^\star\rtimes{\mathfrak g}$, where ${\mathfrak g}$ acts on its dual ${\mathfrak g}^\star$ using the coadjoint action. The metric on $I{\mathfrak g}$ need not ever be explicitly used, yet it is the metric associated with the a norm on $I{\mathfrak g}$, which is simply the contraction map of ${\mathfrak g}^\star$ and ${\mathfrak g}$. This definition appears roughly starting at minute 44:00 of this video. --Drorbn 20:30, 19 March 2012 (EDT)
0:12:50 [add] The tensor map claim.
0:19:44 [add] What is $I{\mathfrak g}$?
0:26:00 [add] What is $I{\mathfrak g}$? (2)
0:28:22 [add] The double is metrized.
0:33:20 [add] The double is metrized (2).
0:36:13 [add] The double is metrized (3).
0:43:27 [add] The double is metrized (4).
1:01:40 [add] The map ${\mathcal T}$.
1:07:59 [add] ${\mathcal T}$ descends to ${\mathcal A}^{wt}$.
1:10:27 [add] ${\mathcal T}$ descends to ${\mathcal A}^{wt}$ (2).
1:16:12 [add] ${\mathcal T}$ descends to ${\mathcal A}^{wt}$ (3).
1:23:04 [add] ${\mathcal T}$ descends to ${\mathcal A}^{wt}$ (4).
1:28:17 [add] The 2D example and ${\mathcal T}$.
1:32:41 [add] The 2D example and ${\mathcal T}$ (2).