In this paper we study quantum del Pezzo surfaces belonging to a certain class. In particular we introduce the generalised Sklyanin-Painlevé algebra and characterise its PBW/PHS/Koszul properties. This algebra contains as limiting cases the generalised Sklyanin algebra, Etingof-Ginzburg and Etingof-Oblomkov-Rains quantum del Pezzo and the quantum monodromy manifolds of the Painlevé equations.
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Acknowledgements. The authors are grateful to Yu. Berest, R. Berger, F. Eshmatov, P. Etingof, D. Gurevich, N. Iyudu, T. Kelly, T. Koornwinder, M. Gross, B. Pym, V. Sokolov, P. Terwilliger, A. Zhedanov for helpful discussions. Our special thanks to Geoffrey Powell who carefully read our first version and made several useful remarks and to the referee whose remarks were truly helpful. This research was supported by the EPSRC Research Grant EP/P021913/1 , by the Hausdorff Institute , by ANR DIADEMS and MPIM (Bonn) and SISSA (Trieste). V.R. was partly supported by the project IPaDEGAN (H2020-MSCA-RISE-2017), Grant Number 778010 , and by the Russian Foundation for Basic Research under the Grants RFBR 18-01-00461 and 16-51-53034-716 GFEN.
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- Calabi-Yau algebras
- Mini-versal deformations
- Painlevé equations
- Sklyanin algebra
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