>John Baez wrote:Very good question! In other words, I haven't the foggiest notion as to
>> 10) Masaki Kashiwara and Yoshihisa Saito, Geometric Construction of
>> Crystal Bases, q-alg/9606009.
>> The "canonical" and "crystal" bases associated to quantum groups,
the answer, and I am dying to know. Lusztig's canonical basis
construction has a rather intimidating reputation, since it uses the
language of "perverse sheaves", which most of us peons haven't gotten
around to mastering. As a result, a lot more people know what the
canonical bases are than *why they really exist*. Answering your
question would require knowing why why they really exist. And that's
why I'm interested in Kashiwara's paper, since it seems to give a
different (and perhaps more comprehensible??) construction.
>In your paper on n-categories you talk about quantisation in categoryThat's right. Ordinarily quantization is conceived of in terms of a
>being a deformation of a symmetric monoidal category to give a braided
>monoidal category and you suggest there may be n-category
>(if I understood you right).
deformation of a commutative algebra to give a noncommutative algebra
(the deformation parameter being Planck's constant). Quantum groups
were a big suprise, and here one has a deformation of a symmetric
monoidal category (the category of representations of your group) to a
braided monoidal category (the category of representations of the
corresponding quantum group). The same sort of thing, in other words,
but one step up the n-categorical ladder. The crystal basis/canonical
basis stuff suggests that there is also a kind of deformation of a
symmetric monoidal 2-category to a braided monoidal 2-category going on!
But this is not just a further prolongation of the same sort of pattern,
since 1) if it were, we would be deforming a strongly involutory
2-category to a weakly involutory one, and 2) as far as I know, the
canonical basis stuff only works when the deformation parameter q is a
root of unity, so the sense in which we have a "deformation" is subtler.
Something very exciting and mysterious is going on here, that's for
>You have also talked about quantisation inThat's not so directly relevant here. You mean a functor from Hilb to
>another context as a functor from a Hilbert space to a larger Hilbert
>space by making a Fock space (that was in categories.html).
Hilb, by the way, taking each space to its Fock space.
>How far have people gone in generalising the idea of forming theVery. Perhaps too.
>category of representations of a group?
>If a representation of aYou mean a representation of a group is a functor from that group
>group is a functor from the group to a Hilbert space can you define
>more general categories of functors from a category?
(regarded as a one-object category) to Hilb. In my paper with Jim Dolan
we stress the idea of a "representation of a category", namely a functor
from that category to Hilb. TQFTs are examples of this, perhaps the
first example that really caught the attention of physicists.
>Is itI don't get what you mean, since we don't use the concept of tensor
>possible to replace tensor products with Cartesian products in this
product in the concept of "functor from a group to Hilb" or "functor from
a category to Hilb". You might be a bit mixed up here.
>Does it make sense to talk about the category of TQFT's onSee above.
>a given cobordism category in a similar way for example?
>I hope these questions make some sense. I don't understand theseWhat you say makes sense, though you seem to be doing a little minor
>castegory concepts very well and I appreciate your potted category
>course in TWF enormously.
level-slipping, speaking of "a Hilbert space" when you probably should
say "Hilb, the category of all Hilbert spaces", and "the category of
TQFTs" when you probably should say "a TQFT". Of course, level-slipping
is to n-category theory as losing your footing is to rock-climbing:
you can't really avoid it when you are pushing your limits. Glad you
like the course.
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