nForum

In a collaborative work at Institute Ruđer Bošković in Zagreb, which will come out as a preprint soon, we are working on a certain noncommutative Hopf algebroid over a noncommutative base coming from Lie algebra theory. I gave a seminar talk about it in mathematics department in Zagreb and got a remark that our approach is too much coordinate based and that it would be desirable to have this work done using coordinate free approach. I came to a one month visit at l’IHÉS and soon realized that in dual language I can tell the story geometrically, though I still do not know how to prove some of the claims without algebraic coordinate-involving stuff which we had before.

So, a Hopf algebroid is made out of a left bialgebroid, right bialgebroid and an antipode map between them.The left and right bialgebroid have the same underlying algebra, but are different as bimodules and corings. I will leave the discussion of the antipode for another post and here sketch how I geometrically arrive at a left bialgebroid.

So let $G$ be a Lie group with Lie algebra $𝔤$ which can be realized as ${𝔤}^L$, the Lie algebra of left invariant vector fields or the Lie algebra ${𝔤}^R$, the Lie algebra of right invariant vector fields; specialization of those vector fields at the unit element $e\in G$ gives the isomorphism of vector spaces with the tangent space $T_{e}G$. The universal enveloping algebra $U(𝔤)$ can be realized also as the algebra of left invariant differential operators $U({𝔤}^L)$ and as $U({𝔤}^R)$. Consider now the algebra $H^L={\mathrm{Diff}}^{\omega }$ of formal differential operators at $e$. That means that you allow finite sums of partial derivatives to any finite order with coefficients which are formal functions, i.e. function supported on an infinitesimal neighborhood of unit element. The usual algebra of (global) differential operators $\mathrm{Diff}(G)$ (which contains $U({𝔤}^L)$ and $U({𝔤}^R)$) is naturally embedded in ${\mathrm{Diff}}^{\omega }$, hence it also contains $U({𝔤}^L)$ and $U({𝔤}^R)$. furthermore the images of the left and right copy of the $U(𝔤)$ mutually commute, what is very important for our story.

Thus we have two maps of $U(𝔤)$ into ${𝒳}^{\omega }$, namely

Here $\alpha$ is defined as extension from the vector fields as a homomorphism of associative algebras, while $\beta$ as antihomomorphism of associative algebras.

If we choose a basis in $𝔤$, i.e. we have a frame in $𝔤$, then we can consider two images in $\Gamma ℱTG$, that is a section of the frame bundle of the tangent bundle: every element of the basis goes into the corresponding left or right invariant vector field. Thus there are also two maps

corresponding to the frames by left and right invariant vector fields. Both maps are analytic and the image is the section of a principal ${\mathrm{GL}}_n$-bundle, hence the difference is a matrix of analytic function on $G$. Better to say, if $\tau$ is the translation map of the principal bundle then $\tau (\alpha (\mathrm{fr}),\beta (\mathrm{fr}))=:{𝒪}^{-1}(\mathrm{fr})$ (for a frame fr) defines some invertible matrix $𝒪$ of analytic functions on $G$. Of course, by the functoriality, this translation matrix does not depend on the choice of frame. We can compute this matrix in a neighborhood of unit element, and I will explain some concrete formula in a later post.

To be continued…

I am interested for many years in using localizations (typically having fully faithful right adjoint) as replacements for open sets in noncommutative algebraic geometry. There are some interesting Čech-like cohomologies lurking there as well (for abelian case known for about 20 years, for nonabelian wait for another paper of mine).

Now I am trying to do, with some ideas from correspondence with colleagues some more new cases, some of which to my surprise are not known. In particular there is lots of issues about descent for functors where both the domain and codomain categories are given by covers by localizations, that is families of localizations such that the induced inverse image functor to the product of the localizations is comonadic. Before tackling that case properly, I am trying to fix some holes in my understanding of the case of endofunctors.

A localization having fully faithful right adjoint is equivalent to a Eilenberg-Moore category over an idempotent monad. For any localization functor, there is a notion of localization compatible endofunctor, which Lunts and Rosenberg introduced in unpublished but available Max Planck Preprint in 1996, in order to study the differential operators on geometries given by “categories of quasicoherent sheaves”. Their notion of compatibility does not involve coherences, and I prefer to work with coherences. I have shown that the coherences actually follow, if one takes for compatibility a distributive law between the idempotent monad $Q$ and the endofunctor $G$, of the form $\mathrm{QG}\Rightarrow \mathrm{GQ}$. More precise statement is at distributive law for idempotent monad (zoranskoda) (cf. also new entry compatible localization with some background references, some on different notions of compatibilities).

The compatibility which is a property in Lunts-Rosenberg becomes a distributive law in my work, what is in general a structure; but a (seamingly new) theorem says that the coherent version is also just a property, namely the distributive law is unique when it exists (“strict compatibility” Edit: It seems to me that the existence of a distributive law is a bit stronger than Lunts-Rosenberg compatibility.) The law is then also invertible what means that its inverse is the other type distributive law $\mathrm{GQ}\Rightarrow \mathrm{QG}$. But what I don’t know is if every other type distributive law $\mathrm{GQ}\Rightarrow \mathrm{QG}$ for an idempotent monad $Q$ is invertible and hence coming from the inverse of a unique distributive law $\mathrm{QG}\Rightarrow \mathrm{GQ}$ or there are genuine non-invertible counterexamples ? Any ideas ?

I’ve moved the nForum ever so slightly sideways. Our web address has changed to:

http://www.math.ntnu.no/~stacey/Mathforge/nForum

The old one will still work - I hope - but all internal links will now point there (those that haven’t been put there by hand), so it ought to “just work” in that if you click on an old link, you get to the page you wanted (but with the wrong URL, and possibly not logged on), but when you click further then you will get to the right page.

I’ll update the links on the nLab when I’m sure that the shift is stable (I can unshift us if necessary).

The reason for the shift is that I want to run a similar forum for the course that I’m teaching, plus there’s the slightly defunct rForum, and John’s new project wants a forum, so I’m consolidating a little. The idea is that they all use the same actual code (rather than copies of the same code) so it’s easier for me to keep them up to date, and they all use the same user database, so that if someone wants to participate in more than one forum here then they don’t have to keep logging in to the different ones (something I was never too happy with as every now and again the cookies got confused).

No doubt I’ve broken something in doing this shift. Please do let me know of any changes in behaviour, particularly with respect to logging in and logging out of the nForum.