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wClips-120314

Speaker: Karene Chu

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Videography by Zsuzsanna Dancso troubleshooting

Notes on wClips-120314:    [edit, refresh]

Section 3.6 - the relation with Lie algebras.


[edit]

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

The wClips

http://katlas.math.toronto.edu/drorbn/dbnvp/dbnvp.png
Date Links
Jan 11, 2012 dbnvp 120111-1: Introduction.
dbnvp 120111-2: Section 2.1 - v-Braids.
Jan 18, 2012 dbnvp 120118-1: An introduction to this web site.
dbnvp 120118-2: Section 2.2 - w-Braids by generators and relations and as flying rings.
dbnvp 120118-3: Section 2.2 - w-Braids - other drawing conventions, "wens".
Jan 25, 2012 dbnvp 120125-1: Section 2.2.3 - basis conjugating automorphisms of F_n.
dbnvp 120125-2: A very quick introduction to finite type invariants in the "u" case.
Feb 1, 2012 dbnvp 120201: 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 dbnvp 120208: Review of u,v, and w braids and of Section 2.3.
Feb 15, 2012 dbnvp 120215: Section 2.5 - mostly compatibilities of Z^w, also injectivity and uniqueness of Z^w.
Feb 22, 2012 dbnvp 120222: 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 dbnvp 120229: 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 dbnvp 120307: Section 3.5: Jacobi diagrams and the bracket-rise theorem.
Mar 14, 2012 dbnvp 120314: Section 3.6 - the relation with Lie algebras.
Mar 21, 2012 dbnvp 120321: Section 4 - Algebraic Structures.
Mar 28, 2012 Out-of-sequence not-on-tape we watched the video of Talks: GWU-1203.
Apr 4, 2012 dbnvp 120404: Section 3.7 - The Alexander Theorem (statement).
Apr 18, 2012 dbnvp 120418: Aside on the Euler trick, the differential of \exp, and the BCH formula.
Apr 25, 2012 dbnvp 120425: Section 3.8, a disorganized lecture towards the proof of the Alexander theorem.
May 2, 2012 dbnvp 120502: Section 4: Algebraic structures (review), circuit algebras, v- and w-tangles.
May 10, 2012 dbnvp 120510: Sections 5.1 and 5.2: tangles, their projectivization and its relationship with Alekseev-Torossian spaces.
May 23, 2012 dbnvp 120523: Section 5.2: Proof of the relationship with A-T spaces.
May 30, 2012 dbnvp 120530: Interpreting {\mathcal A}^w(\uparrow_n) as a universal space of invariant tangential differential operators.
wClips Seminar Group Photo
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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Managed by dbnvp: Click the "h:mm:ss" links on the right panel to jump to a specific video time.

0:00:00 [edit] Blackboard shots by Zsuzsanna Dancso.

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

0:04:47 [edit] 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 [edit] 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).
0:47:43 [add] The co-commutative case.
0:54:22 [add] The 2D example.
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).