Quantised Painlev\'e monodromy manifolds, Sklyanin and Calabi-Yau algebras
Leonid Chekhov, Marta Mazzocco, Volodya Rubtsov
TL;DR
This paper introduces a new algebraic framework for quantum del Pezzo surfaces, connecting Sklyanin, Painlevé, and Calabi-Yau structures, and analyzes their algebraic properties.
Contribution
It defines the generalized Sklyanin-Painlevé algebra and characterizes its PBW, PHS, and Koszul properties, unifying several quantum geometric structures.
Findings
The algebra encompasses various known quantum del Pezzo and Painlevé monodromy manifolds.
It establishes the algebra's PBW, PHS, and Koszul properties.
Provides a new algebraic approach to quantum geometric surfaces.
Abstract
In this paper we study quantum del Pezzo surfaces belonging to a certain class. In particular we introduce the generalised Sklyanin-Painlev\'e 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 Painlev\'e equations.
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Taxonomy
TopicsAlgebraic structures and combinatorial models · Nonlinear Waves and Solitons · Advanced Topics in Algebra