# History of the Odd Dimensional Quantum Spheres

After reading this question, I began to wonder about the history of quantum $(2N-1)$-spheres. Basically I have two questions:

(1) Who first introduced the $(2N-1)$-spheres, and who first introduced them as invariant subalgebras of quantum $SU_N$? I know that Podles is often credited with introducing quantum spheres, but as far as I can see (his paper is difficult to find on the web) he only introduced a family of deformations of $2$-spheres, which are not of course of the form $S^{2N-1}_q$. I know also that something appeared in the early FRT-papers, but these are also difficult to find, and are in Russian.

(2) The proof in Klimyk and Schmudgen of the generators and relations result is based on a 1993 paper by Noumi, Yamada, and Mimachi. (This is also difficult to find.) Was this the first generators and relations description for the for $S^{2N-1}_q$? Did it generalise an earlier result, ie for $S^5_q$?

P.S. I would also be very interested in hearing about the history of quantum projective spaces. Who first introduced them? Who first introduced them as invariant subalgebras of quantum $SU_N$? Who first introduced them as invariant subalgebras of the quantum spheres?

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(1) I think Podles only introduced the quantum 2-spheres. His paper is linked from MathSciNet, so you should be able to get it if you have access to ams.org. I think the higher-dimensional spheres were first considered by Vaksman and Soibelman in http://www.ams.org/mathscinet-getitem?mr=1086447. Unfortunately that paper, while it was translated into English, appears not to be available online.

(2) Not sure, sorry.

About the projective spaces: I think http://www.ams.org/mathscinet-getitem?mr=1432197, by Dijkhuizen and Noumi, is one of the early papers, but they refer also to http://www.ams.org/mathscinet-getitem?mr=1183477 by Korogodsky and Vaksman, which I haven't really looked at but seems to contain some relevant stuff. And really the paper by Noumi et al is quite early. I'm not sure about who first thought about them as invariant subalgebras of the quantum spheres.

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I should note also that most of this info came from the notes at the end of Chapter 11 of Klimyk and Schmudgen. They aren't exhaustive, but they do cover a lot of references in those end-notes. –  MTS Mar 7 '11 at 20:56