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A quadratic Poisson Gel'fand-Kirillov problem in prime characteristic


Authors: Stéphane Launois and César Lecoutre
Journal: Trans. Amer. Math. Soc. 368 (2016), 755-785
MSC (2010): Primary 17B63, 20G42
Published electronically: May 13, 2015
MathSciNet review: 3430349
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Abstract: The quadratic Poisson Gel'fand-Kirillov problem asks whether the field of fractions of a Poisson algebra is Poisson birationally equivalent to a Poisson affine space, i.e. to a polynomial algebra $ \mathbb{K}[X_1, \dots , X_n]$ with Poisson bracket defined by $ \{X_i,X_j\}=\lambda _{ij} X_iX_j$ for some skew-symmetric matrix $ (\lambda _{ij}) \in M_n(\mathbb{K})$. This problem was studied in 2011 by Goodearl and Launois over a field of characteristic 0 by using a Poisson version of the deleting derivation homomorphism of Cauchon. In this paper, we study the quadratic Poisson Gel'fand-Kirillov problem over a field of arbitrary characteristic. In particular, we prove that the quadratic Poisson Gel'fand-Kirillov problem is satisfied for a large class of Poisson algebras arising as semiclassical limits of quantised coordinate rings. We introduce the concept of higher Poisson derivation, which allows us to extend the Poisson version of the deleting derivation homomorphism from the characteristic 0 case to the case of arbitrary characteristic.

When a torus is acting rationally by Poisson automorphisms on a Poisson polynomial algebra arising as the semiclassical limit of a quantised coordinate ring, we prove (under some technical assumptions) that quotients by Poisson prime torus-invariant ideals also satisfy the quadratic Poisson Gel'fand-Kirillov problem. In particular, we show that coordinate rings of determinantal varieties satisfy the quadratic Poisson Gel'fand-Kirillov problem.


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Additional Information

Stéphane Launois
Affiliation: School of Mathematics, Statistics and Actuarial Science (SMSAS), Cornwallis Building, University of Kent, Canterbury, Kent CT2 7NF, United Kingdom
Email: S.Launois@kent.ac.uk

César Lecoutre
Affiliation: School of Mathematics, Statistics and Actuarial Science (SMSAS), Cornwallis Building, University of Kent, Canterbury, Kent CT2 7NF, United Kingdom
Email: cl335@kent.ac.uk

DOI: https://doi.org/10.1090/tran/6352
Keywords: Poisson algebra, Gel'fand-Kirillov conjecture, prime characteristic
Received by editor(s): February 8, 2013
Received by editor(s) in revised form: December 2, 2013
Published electronically: May 13, 2015
Additional Notes: The first author is grateful for the financial support of EPSRC first grant EP/I018549/1.
The second author thanks EPSRC for its support
Article copyright: © Copyright 2015 American Mathematical Society