(no subject)

[leblon]

А вот такой есть интересный вопрос. Категорию конечномерных представлений группы Ли можно продеформировать в сплетенную моноидальную, получив категорию представлений квантовой группы. С другой стороны, можно рассматривать категорию конечномерных представлений как категорию эквивариантных когерентных пучков на точке, и тогда естественно задать вопрос: для каких комплексных (или даже аффинных) многообразий с действием комплексной (или даже аффинной) группы Ли эквивариантная (производная) категория когерентных пучков допускает подобную деформацию? Я поначалу думал, что для этого многообразие должно быть симплектическим, а действие группы - гамильтоновым. Но теперь мне кажется, что это не так. Например, кажется, что в качестве многообразия можно взять алгебру Ли самой группы (ну, это еще куда ни шло, на ней хоть пуассонова структура есть), или даже саму группу (а это уже совсем странно), с присоединенным действием.

Comments

[hippie57.livejournal.com]

People have thought about this. Then the question becomes as follows: how to realize the derived category of modules over a quantum group in terms of coherent sheaves somewhere? I think that a possible geometric object is closely related to G/G, but the object itself depends on the parameter of quantum deformation, for example a possible answer could have the form (a G-variety, a gerbe on it). So coherent sheaves could be in fact twisted coherent sheaves. As you know partially the answer is given by T^*G/B for the Langlands dual group.

[leblon.livejournal.com]

Why is it the same question? I do not want to realize the same old category of modules over the quantum group geometrically; I want to find new examples of braided monoidal deformations of symmetric tensor categories. I know for a fact that there is such a deformation for the G-equivariant derived category of coherent sheaves on G, where G acts on itself by conjugation.

[hippie57.livejournal.com]

Well, a possible explanation for the Freed-Hopkins-Teleman business could be as follows: 1) one wants to realize geometrically not the Verlinde category but the whole category of G[[s]]-integrable g((s))-modules at a positive level. 2) This category (a certain reasonable subcategory in the derived category) is equivalent to a similar category at the negative level. By Kazhdan-Lusztig the latter one is equivalent to the category of modules over the Lusztig quantum group at the corresponding root of unity. The Verlinde category is a (monoidal) quotient of this category. 3) The rest is just a hope. Namely, that there would be a category of twisted coherent sheaves on G^\vee/G^\vee (monoidally) equivalent to the derived category of modules over the quantum group.

Btw, how do you deform the monoidal category of sheaves on G/G? Is the monoidal structure given by convolution or by tensor product?

[leblon.livejournal.com]

Your question deconfused me. Sometimes some wires cross in my head and I confuse the convolution monoidal structure with the tensor product one. This probably happens because in my paper both appear, in slightly different circumstances (depending on which object in the 2-category of boundary conditions in the 3d TFT I consider, I may get either convolution or tensor product monoidal structures). In the case of G/G I need to start with the convolution monoidal structure in order to be able to deform it. On the other hand, if I take instead of G some complex symplectic manifold X with a hamiltonian G-action, then the tensor product on the G-equivariant derived category of coherent sheaves on X can be deformed into a braided monoidal structure. Problem resolved!

[hippie57.livejournal.com]

It could be that Langlands duality interchanges the (deformed) convolution tensor product and the (deformed) coherent sheaf one. This is ot easy to imagine, but on the other hand, if the duality is of Fourier transform flavour, this could happen.

[hippie57.livejournal.com]

E.g. that's what happens for G=T, the torus.