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    • CommentRowNumber1.
    • CommentAuthorBruce
    • CommentTimeApr 18th 2010
    I started a page titled <a href="http://www.ncatlab.org/nlab/show/Help+me!+I%27m+trying+to+understand+Bakalov+and+Kirillov">Help me! I'm trying to understand Bakalov and Kirillov</a> for those (like me) who need help in understanding some of the calculations in this book. I posted my first question. Will try to ask people to answer it.
    • CommentRowNumber2.
    • CommentAuthorAndrew Stacey
    • CommentTimeApr 19th 2010

    Link: Help me! I’m trying to understand Bakalov and Kirillov

    If I may borrow a parlance from mathoverflow: +1 for the descriptive title, Bruce!

    • CommentRowNumber3.
    • CommentAuthorEric
    • CommentTimeApr 19th 2010

    This is an example that I think would make more sense on this forum than the nLab. It is more of a discussion. Once you understand Bakalov and Kirillov, then we can transfer that newly found knowledge to the nLab.

    We just need to figure out a way to attract eyes over here instead.

    • CommentRowNumber4.
    • CommentAuthorEric
    • CommentTimeApr 19th 2010

    From the nLab:

    How does the “s” map work?

    Bruce Bartlett: I don’t understand the ss map in Example 5.1.11 in the online version of the book. It’s supposed to be the automorphism

    s:Γ 1,1Γ 1,1 s : \Gamma_{1,1} \rightarrow \Gamma_{1,1}

    of the torus with one boundary circle which is the analogue of a map which I do understand, the automorphism

    s t:Γ 1,0Γ 1,0 s_t : \Gamma_{1,0} \rightarrow \Gamma_{1,0}

    of the ordinary torus (no boundary circle) to itself, which sends

    (θ,ϕ)(ϕ,θ). (\theta, \phi) \mapsto (\phi, -\theta).

    But the map s ts_t doesn’t seem to work if there is a boundary circle involved. Because an automorphism of Γ 1,1\Gamma_{1,1} is supposed to be the identity on the boundary circle, but the map s ts_t isn’t: it rotates the boundary circle a quarter rotation (right?). The reason I say so is the following picture: we think of the torus as 2/ 2\mathbb{R}^2 / \mathbb{Z}^2, as Bakalov and Kirillov encourage us to do. Then the presence of the boundary circle can be thought of as having a little tangent vector pointing to the right at all the lattice positions (this uses their alternative Definition 5.1.10 the extended surface category). The map (x,y)(y,x)(x,y) \mapsto (y, -x) rotates the plane by a quarter rotation and takes the lattice to itself, hence it descends to the quotient. But this map rotates the tangent vector by a quarter rotation, and we’re supposed to fix the tangent vectors!

    So how does the map ss work?

  1. Eric,

    the map ss should be precisely as you describe it, but you keep fixes the S 1S^1. see it this way: realize your torus with boundary circle as a rubber square with identifications on the opposite sides, with a circular hole in the middle. then turn the square of a 1/4 rotation, while keeping the boundary of the hole pointwise fixed (the homotopy class of this map is well defined).

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeApr 19th 2010
    • (edited Apr 19th 2010)

    This is an example that I think would make more sense on this forum than the nLab. It is more of a discussion. Once you understand Bakalov and Kirillov, then we can transfer that newly found knowledge to the nLab.

    I second that.

    At least I suggest the following: if you do want such discussion to be had on an nLab page, create a page with a title that will be the correct title after the entry has stabilized. Here for instance it might be Bakalov-Kirillov TFT or something like that. Then create in that entry a section “Discussion”.

    Because I think we should have in mind the usability of the nLab in, say, five yearss from when a given entry is created. Probably an entry titled “Help me understand xyz” is completely obsolete then, because somebody will have helped, somebody will have understood and some stable material will have been written up elsewhere.

    • CommentRowNumber7.
    • CommentAuthorBruce
    • CommentTimeApr 19th 2010
    • (edited Apr 19th 2010)
    I'm surprised you think this is better suited to the nForum than the nLab. I think "question and answer" type pages for textbooks would be very useful on the nLab. I can imagine a "Help me! I'm trying to understand Milnor's 'Morse Theory'" page. But ok, too bad.

    Domenico: Thanks. I was initially thinking that way, but then I confused myself with the following picture: Try to lift this automorphism to the level of the plane. In other words, imagine the plane with little boundary circles at each lattice point in Z^2. We want to perform some kind of "stretched rotation" on the plane which descends to the s-map on the quotient. That confused me because we need to do that "stretching" at each boundary circle (not just at the one at the origin). But now I guess it's possible: we ask a heavy person to sit on a part of the plane where there are no boundary circles ("knobs"), so as to keep the plane (thought of as an "elastic carpet") fixed. Then we rotate each of the knobs clockwise a quarter rotation, stretching the carpet as it goes. This procedure is invariant under the lattice symmetry, so it descends to the torus (but I can't imagine it at the level of the plane, it must be complicated, because turning the knob here will affect the carpet near the other knobs...). Am I still confused?
    • CommentRowNumber8.
    • CommentAuthorBruce
    • CommentTimeApr 19th 2010
    Domenico, not Domenica! (sincere apologies)
    • CommentRowNumber9.
    • CommentAuthordomenico_fiorenza
    • CommentTimeApr 19th 2010
    • (edited Apr 19th 2010)

    at the level of the plane, you can think it as a two-steps process: first, you turn all S 1S^1’s of a 1/4 clockwise (this is not an automorphism of your structure, since it moves a basepoint in the S 1S^1); then turn all the plane of a 1/4 rotation anticlockwise around the origin. the composition of these two is the transformation you’re interested in.

    (no need to apologize :-) )

    • CommentRowNumber10.
    • CommentAuthorBruce
    • CommentTimeApr 19th 2010
    Thanks, yes, I forgot to add that last part of turning the plane 1/4 of a rotation back afterwards. I'm busy trying to draw a diagram for myself where I connect up the little circles in the plane with horizontal and vertical lines and then perform that rotation continuously to see what happens to those lines... it looks like they become horizontal and vertical "integral signs". Ok, I'm happy now, thanks!
    • CommentRowNumber11.
    • CommentAuthorBruce
    • CommentTimeApr 19th 2010
    I adjusted the nLab page to make it clearer what the algorithm is: get your question answered at the nForum or MathOverflow, then type up the answer when you're happy with it in the nLab.
    • CommentRowNumber12.
    • CommentAuthorzskoda
    • CommentTimeApr 19th 2010

    I somewhat disagree with 6. I mean, my past experience is that it is pretty hard to use mixed pages with blocks of normal content with large pages-long sections of discussion. Discussion is useful only in certain mode and having lots of it in regular page is kind of distracting from seeing the main part. I mean, the pages with more work character like informal write ups of seminars, which appeared before in nlab should stay I think in such a format, and concept pages should be created as more clean pages separately. Thus I would not like the learning discussion being a part of future page on some approach to modular functors and CFT but rather a separate accompanying page; thus a variant of the present page title like "Journal club on Bakalov-Kirillov book" is good enough for me; while both the general reference/book entry with toc as well as entries for concepts/topics partly covered in that book like modular functor, Kirillov-Bakalov RCFT etc. should also be separated. Thus I advocate not only making different pages according to different topics as Urs does but also with respect to different purposes.

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeApr 19th 2010

    Sure, whatever works. Whatever we do, let’s make sure that pages become such that they can actually be used as decent sources of information after the dust has settled and the activity has gone elsewhere.

    • CommentRowNumber14.
    • CommentAuthorzskoda
    • CommentTimeApr 19th 2010

    Right, long term purpose is important. If it were just for daily usage, nlab would not be anything special...

    • CommentRowNumber15.
    • CommentAuthorBruce
    • CommentTimeApr 21st 2010
    Here's another question: I can't verify the statement

    "It is easy to check that the operator S defined by (3.1.32) satisfies (5.5.3)"

    in the -online- version of Bakalov and Kirillov, on page 118.

    When I substitute the s operator in, the LHS and RHS aren't topologically equivalent!

    (Heh I' guess no-one will ever answer this technical question...)
  2. don’t know.. in the published version it is correct (and a sequence of moves showing the equivalence of RHS and LHS is showed)

    • CommentRowNumber17.
    • CommentAuthorBruce
    • CommentTimeApr 27th 2010
    Thanks a lot Domenico. Sorry for having taken so long to notice this reply. I will get a hold of the book version.
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