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arXiv · 1111.6044

Solution of a q-difference Noether problem and the quantum Gelfand-Kirillov conjecture for gl_N

Vyacheslav Futorny·Jonas T. Hartwig
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Abstract

It is shown that the q-difference Noether problem for all classical Weyl groups has a positive solution, simultaneously generalizing well known results on multisymmetric functions of Mattuck and Miyata in the case q=1, and q-deforming the noncommutative Noether problem for the symmetric group. It is also shown that the quantum Gelfand-Kirillov conjecture for gl_N (for a generic q) follows from the positive solution of the q-difference Noether problem for the Weyl group of type D_n. The proof is based on the theory of Galois rings developed by the first author and Ovsienko. From here we obtain a new proof of the quantum Gelfand-Kirillov conjecture for sl_N, thus recovering the result of Fauquant-Millet. Moreover, we provide an explicit description of skew fields of fractions for quantized gl_N and sl_N generalizing Alev and Dumas.

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Vyacheslav Futorny, Jonas T. Hartwig. 2011-11-29. Solution of a q-difference Noether problem and the quantum Gelfand-Kirillov conjecture for gl_N. https://doi.org/10.1007/s00209-013-1184-3

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