Multiplicative quiver varieties and generalised Ruijsenaars-Schneider models

Abstract

We study some classical integrable systems naturally associated with multiplicative quiver varieties for the (extended) cyclic quiver with $m$ vertices. The phase space of our integrable systems is obtained by quasi-Hamiltonian reduction from the space of representations of the quiver. Three families of Poisson-commuting functions are constructed and written explicitly in suitable Darboux coordinates. The case $m=1$ corresponds to the tadpole quiver and the Ruijsenaars-Schneider system and its variants, while for $m>1$ we obtain new integrable systems that generalise the Ruijsenaars-Schneider system. These systems and their quantum versions also appeared recently in the context of supersymmetric gauge theory and cyclotomic DAHAs, as well as in the context of the Macdonald theory.

27 pages

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Cited by in corpus (28)

• Poisson-Lie analogues of spin Sutherland models
• On the Hamiltonian formulation of the trigonometric spin Ruijsenaars-Schneider system
• Commutative families in DIM algebra, integrable many-body systems and $q,t$ matrix models
• Spin versions of the complex trigonometric Ruijsenaars-Schneider model from cyclic quivers
• Multiplicative preprojective algebras are 2-Calabi-Yau
• Morphisms of double (quasi-)Poisson algebras and action-angle duality of integrable systems
• Superintegrability of Calogero-Moser systems associated with the cyclic quiver
• Hamiltonian structures for integrable nonabelian difference equations
• On Chalykh's approach to eigenfunctions of DIM-induced integrable Hamiltonians
• Calabi-Yau structures for multiplicative preprojective algebras
• Chalykh's Baker-Akhiezer functions as eigenfunctions of the integer-ray integrable systems
• Double quasi-Poisson brackets : fusion and new examples
• Classical $N$-Reflection Equation and Gaudin Models
• Reduction of a bi-Hamiltonian hierarchy on $T^{\ast }\mathrm{U}(n)$ to spin Ruijsenaars--Sutherland models
• Self-duality and scattering map for the hyperbolic van Diejen systems with two coupling parameters (with an appendix by S. Ruijsenaars)
• Integrable multi-Hamiltonian systems from reduction of an extended quasi-Poisson double of $U(n)$
• Integrable systems on multiplicative quiver varieties from cyclic quivers
• On the noncommutative Poisson geometry of certain wild character varieties
• CMM formula as superintegrability property of unitary model
• Bi-Hamiltonian structure of Sutherland models coupled to two $\mathfrak{u}(n{)}^{\ast }$-valued spins from Poisson reduction
• Generating twisted Cherednik eigenfunctions
• Application of the reflection functor to integrable systems on quiver varieties
• Elliptic triad
• Diamond of triads
• Symmetric polynomials: DIM integrable systems versus twisted Cherednik systems
• Twisted Cherednik spectrum as a $q,t$-deformation
• A basic triad in Macdonald theory
• Twisted Baker-Akhiezer function from determinants