I'm doing my bachelor thesis on explaining an article of Cattaneo, Felder and Tomassini, and here is the link to the article: https://arxiv.org/pdf/math/0012228.pdf. I'm currently working on the page 4. Please, read it shortly for the next. I encounter a problem in the last section of this page, in order to undestand what is meant:
I had first difficulties in order to understand what was meant by "the Maurer-Cartan form is $GL(d,R)$-equivariant". I tried to searched the signification of the last term, but although I found several papers saying that something was $GL(d,R)$-equivariant, I couldn't find any definition of it. I supposed a moment that it could mean that the Maurer-Cartan form was invariant under pulling-back by the right action of $GL(d,R)$ on $M^{coor}$ but after some calcultations, I have seen that it was wrong. Do you have an idea of the significance of it, or could you give me a hint about where I could find that?
The second thing was about the $M^{coor}×_{GL(d,R)}R[[y^1,...,y^d]]$. I thought that it was a fibre product like I did it in the lecture but, I found several problems by considering it so. First of all, the article does not define any maps $M^{coor}→GL(d,R)$ and $R[[y^1,...,y^d]]→GL(d,R)$ which are necessary in order to define the fibre product and I don't see any canonical maps for it. Second thing: I beleave that $R[[y^1,...,y^d]]$ is not a fiber bundle over $GL(d,R)$ (It would have been one over the set of all the dxd matrices but not if you restrict it to $GL(d,R))$. Last problem: It would then define a fibre bundle over $M^{coor}$ and not over $M^{aff}$ like it is said in the article. Am I right, when saying that it is not a "classical" fibre bundle? If yes, do you know how I should understand it?
Thank you very much for your help!