We construct quantizations of multiplicative hypertoric varieties using an algebra of q-difference operators on affine space, where q is a root of unity in C. The quantization defines a matrix bundle (i.e. Azumaya algebra) over the multiplicative hypertoric variety and admits an explicit finite étale splitting. The global sections of this Azumaya algebra is a hypertoric quantum group, and we prove a localization theorem. We introduce a general framework of Frobenius quantum moment maps and their Hamiltonian reductions; our results shed light on an instance of this framework.
Journal of Algebra
National Science Foundation: Graduate Research Fellowship and grant No.0932078000; ERC Advanced Grant “Arithmetic and Physics of Higgs moduli spaces” No. 320593 The author is grateful to David Jordan for suggesting this project and providing guidance throughout, particularly for the formulation of Frobenius quantum moment maps and key ideas in the proofs of Theorems 3.12 and 4.8. Special thanks to David Ben-Zvi (the author's PhD advisor) for numerous discussions and constant encouragement, and for suggesting the term ‘hypertoric quantum group.’ Many results appearing in the current paper were proven independently by Nicholas Cooney; the author is grateful to Nicholas for sharing his insight on various topics, including Proposition 3.8. The author also thanks Nicholas Proudfoot for relating the definition of multiplicative hypertoric varieties, as well as the content of Remark 2.14. The author also benefited immensely from the close reading and detailed comments of an anonymous referee, and from conversations with Justin Hilburn, Kobi Kremnitzer, Michael McBreen, Tom Nevins, Travis Schedler, and Ben Webster.
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Ganev IV. Quantizations of multiplicative hypertoric varieties at a root of unity. Journal of Algebra. 2018;506:92-128. doi:10.1016/j.jalgebra.2018.03.015
Ganev, I. V. (2018). Quantizations of multiplicative hypertoric varieties at a root of unity. Journal of Algebra. World Scientific Publishing. https://doi.org/10.1016/j.jalgebra.2018.03.015
Ganev, Iordan V. “Quantizations of Multiplicative Hypertoric Varieties at a Root of Unity.” Journal of Algebra. World Scientific Publishing, 2018. https://doi.org/10.1016/j.jalgebra.2018.03.015.
I. V. Ganev, “Quantizations of multiplicative hypertoric varieties at a root of unity,” Journal of Algebra, vol. 506. World Scientific Publishing, pp. 92–128, 2018.
Ganev IV. 2018. Quantizations of multiplicative hypertoric varieties at a root of unity. Journal of Algebra. 506, 92–128.
Ganev, Iordan V. “Quantizations of Multiplicative Hypertoric Varieties at a Root of Unity.” Journal of Algebra, vol. 506, World Scientific Publishing, 2018, pp. 92–128, doi:10.1016/j.jalgebra.2018.03.015.