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Journal articles on the topic 'Calogero-Moser spaces'

Author: Grafiati

Published: 5 October 2024

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1

Bellamy, Gwyn. "On singular Calogero-Moser spaces." Bulletin of the London Mathematical Society 41, no. 2 (March 11, 2009): 315–26. http://dx.doi.org/10.1112/blms/bdp019.

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2

BEREST, YURI, ALIMJON ESHMATOV, and FARKHOD ESHMATOV. "MULTITRANSITIVITY OF CALOGERO-MOSER SPACES." Transformation Groups 21, no. 1 (August 19, 2015): 35–50. http://dx.doi.org/10.1007/s00031-015-9332-y.

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3

Ben-Zvi, David, and Thomas Nevins. "Perverse bundles and Calogero–Moser spaces." Compositio Mathematica 144, no. 6 (November 2008): 1403–28. http://dx.doi.org/10.1112/s0010437x0800359x.

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AbstractWe present a simple description of moduli spaces of torsion-free 𝒟-modules (𝒟-bundles) on general smooth complex curves, generalizing the identification of the space of ideals in the Weyl algebra with Calogero–Moser quiver varieties. Namely, we show that the moduli of 𝒟-bundles form twisted cotangent bundles to moduli of torsion sheaves on X, answering a question of Ginzburg. The corresponding (untwisted) cotangent bundles are identified with moduli of perverse vector bundles on T*X, which contain as open subsets the moduli of framed torsion-free sheaves (the Hilbert schemes T*X[n] in the rank-one case). The proof is based on the description of the derived category of 𝒟-modules on X by a noncommutative version of the Beilinson transform on P1.

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4

Berest, Yuri. "Calogero–Moser spaces over algebraic curves." Selecta Mathematica 14, no. 3-4 (March 21, 2009): 373–96. http://dx.doi.org/10.1007/s00029-009-0518-9.

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5

Kuyumzhiyan, Karine. "Infinite transitivity for Calogero-Moser spaces." Proceedings of the American Mathematical Society 148, no. 9 (June 8, 2020): 3723–31. http://dx.doi.org/10.1090/proc/15030.

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6

Bellamy, Gwyn. "Factorization in generalized Calogero–Moser spaces." Journal of Algebra 321, no. 1 (January 2009): 338–44. http://dx.doi.org/10.1016/j.jalgebra.2008.09.015.

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7

Andrist, Rafael. "The density property for Calogero–Moser spaces." Proceedings of the American Mathematical Society 149, no. 10 (July 2, 2021): 4207–18. http://dx.doi.org/10.1090/proc/15457.

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We prove the algebraic density property for the Calogero–Moser spaces C n {\mathcal {C}_{n}} , and give a description of the identity component of the group of holomorphic automorphisms of C n {\mathcal {C}_{n}} .

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8

HAINE, LUC, EMIL HOROZOV, and PLAMEN ILIEV. "TRIGONOMETRIC DARBOUX TRANSFORMATIONS AND CALOGERO–MOSER MATRICES." Glasgow Mathematical Journal 51, A (February 2009): 95–106. http://dx.doi.org/10.1017/s0017089508004813.

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AbstractWe characterize in terms of Darboux transformations the spaces in the Segal–Wilson rational Grassmannian, which lead to commutative rings of differential operators having coefficients which are rational functions of ex. The resulting subgrassmannian is parametrized in terms of trigonometric Calogero–Moser matrices.

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9

Oblomkov, Alexei. "Double affine Hecke algebras and Calogero-Moser spaces." Representation Theory of the American Mathematical Society 8, no. 10 (June 2, 2004): 243–66. http://dx.doi.org/10.1090/s1088-4165-04-00246-8.

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10

Horozov, Emil. "Calogero-Moser spaces and an adelic $W$-algebra." Annales de l’institut Fourier 55, no. 6 (2005): 2069–90. http://dx.doi.org/10.5802/aif.2152.

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11

Bellamy, Gwyn, and Victor Ginzburg. "SL2-action on Hilbert Schemes and Calogero-Moser spaces." Michigan Mathematical Journal 66, no. 3 (August 2017): 519–32. http://dx.doi.org/10.1307/mmj/1496995337.

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12

Eshmatov, Farkhod, Alimjon Eshmatov, and Yuri Berest. "On subgroups of the Dixmier group and Calogero-Moser spaces." Electronic Research Announcements in Mathematical Sciences 18 (March 2011): 12–21. http://dx.doi.org/10.3934/era.2011.18.12.

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13

Chabaud, Ulysse, and Saeed Mehraban. "Holomorphic representation of quantum computations." Quantum 6 (October 6, 2022): 831. http://dx.doi.org/10.22331/q-2022-10-06-831.

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We study bosonic quantum computations using the Segal-Bargmann representation of quantum states. We argue that this holomorphic representation is a natural one which not only gives a canonical description of bosonic quantum computing using basic elements of complex analysis but also provides a unifying picture which delineates the boundary between discrete- and continuous-variable quantum information theory. Using this representation, we show that the evolution of a single bosonic mode under a Gaussian Hamiltonian can be described as an integrable dynamical system of classical Calogero-Moser particles corresponding to the zeros of the holomorphic function, together with a conformal evolution of Gaussian parameters. We explain that the Calogero-Moser dynamics is due to unique features of bosonic Hilbert spaces such as squeezing. We then generalize the properties of this holomorphic representation to the multimode case, deriving a non-Gaussian hierarchy of quantum states and relating entanglement to factorization properties of holomorphic functions. Finally, we apply this formalism to discrete- and continuous- variable quantum measurements and obtain a classification of subuniversal models that are generalizations of Boson Sampling and Gaussian quantum computing.

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14

Przeździecki, Tomasz. "The combinatorics of C⁎-fixed points in generalized Calogero-Moser spaces and Hilbert schemes." Journal of Algebra 556 (August 2020): 936–92. http://dx.doi.org/10.1016/j.jalgebra.2020.04.003.

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15

Kesten, J., S. Mathers, and Z. Normatov. "Infinite transitivity on the Calogero-Moser space C2." Algebra and Discrete Mathematics 31, no. 2 (2021): 227–50. http://dx.doi.org/10.12958/adm1656.

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We prove a particular case of the conjecture of Berest--Eshmatov--Eshmatov by showing that the group of unimodular automorphisms of C[x,y] acts in an infinitely-transitive way on the Calogero-Moser space C2.

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16

Prykarpatski, Anatolij K. "Quantum Current Algebra in Action: Linearization, Integrability of Classical and Factorization of Quantum Nonlinear Dynamical Systems." Universe 8, no. 5 (May 20, 2022): 288. http://dx.doi.org/10.3390/universe8050288.

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This review is devoted to the universal algebraic and geometric properties of the non-relativistic quantum current algebra symmetry and to their representations subject to applications in describing geometrical and analytical properties of quantum and classical integrable Hamiltonian systems of theoretical and mathematical physics. The Fock space, the non-relativistic quantum current algebra symmetry and its cyclic representations on separable Hilbert spaces are reviewed and described in detail. The unitary current algebra family of operators and generating functional equations are described. A generating functional method to constructing irreducible current algebra representations is reviewed, and the ergodicity of the corresponding representation Hilbert space measure is mentioned. The algebraic properties of the so called coherent states are also reviewed, generated by cyclic representations of the Heisenberg algebra on Hilbert spaces. Unbelievable and impressive applications of coherent states to the theory of nonlinear dynamical systems on Hilbert spaces are described, along with their linearization and integrability. Moreover, we present a further development of these results within the modern Lie-algebraic approach to nonlinear dynamical systems on Poissonian functional manifolds, which proved to be both unexpected and important for the classification of integrable Hamiltonian flows on Hilbert spaces. The quantum current Lie algebra symmetry properties and their functional representations, interpreted as a universal algebraic structure of symmetries of completely integrable nonlinear dynamical systems of theoretical and mathematical physics on functional manifolds, are analyzed in detail. Based on the current algebra symmetry structure and their functional representations, an effective integrability criterion is formulated for a wide class of completely integrable Hamiltonian systems on functional manifolds. The related algebraic structure of the Poissonian operators and an effective algorithm of their analytical construction are described. The current algebra representations in separable Hilbert spaces and the factorized structure of quantum integrable many-particle Hamiltonian systems are reviewed. The related current algebra-based Hamiltonian reconstruction of the many-particle oscillatory and Calogero–Moser–Sutherland quantum models are reviewed and discussed in detail. The related quasi-classical quantum current algebra density representations and the collective variable approach in equilibrium statistical physics are reviewed. In addition, the classical Wigner type current algebra representation and its application to non-equilibrium classical statistical mechanics are described, and the construction of the Lie–Poisson structure on the phase space of the infinite hierarchy of distribution functions is presented. The related Boltzmann–Bogolubov type kinetic equation for the generating functional of many-particle distribution functions is constructed, and the invariant reduction scheme, compatible with imposed correlation functions constraints, is suggested and analyzed in detail. We also review current algebra functional representations and their geometric structure subject to the analytical description of quasi-stationary hydrodynamic flows and their magneto-hydrodynamic generalizations. A unified geometric description of the ideal idiabatic liquid dynamics is presented, and its Hamiltonian structure is analyzed. A special chapter of the review is devoted to recent results on the description of modified current Lie algebra symmetries on torus and their Lie-algebraic structures, related to integrable so-called heavenly type spatially many-dimensional dynamical systems on functional manifolds.

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17

BONNAFÉ, CÉDRIC. "AUTOMORPHISMS AND SYMPLECTIC LEAVES OF CALOGERO–MOSER SPACES." Journal of the Australian Mathematical Society, October 17, 2022, 1–32. http://dx.doi.org/10.1017/s1446788722000180.

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Abstract We study the symplectic leaves of the subvariety of fixed points of an automorphism of a Calogero–Moser space induced by an element of finite order of the normalizer of the associated complex reflection group. We give a parametrization à la Harish-Chandra of its symplectic leaves (generalizing earlier works of Bellamy and Losev). This result is inspired by the mysterious relations between the geometry of Calogero–Moser spaces and unipotent representations of finite reductive groups, which is the theme of another paper, C. Bonnafé [‘Calogero–Moser spaces vs unipotent representations’, Pure Appl. Math. Q., to appear, Preprint, 2021, arXiv:2112.13684].

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18

Bonnafé, Cédric, and Ulrich Thiel. "Computational aspects of Calogero–Moser spaces." Selecta Mathematica 29, no. 5 (October 24, 2023). http://dx.doi.org/10.1007/s00029-023-00878-3.

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AbstractWe present a series of algorithms for computing geometric and representation-theoretic invariants of Calogero–Moser spaces and rational Cherednik algebras associated with complex reflection groups. In particular, we are concerned with Calogero–Moser families (which correspond to the $$\mathbb {C}^\times $$ C × -fixed points of the Calogero–Moser space) and cellular characters (a proposed generalization by Rouquier and the first author of Lusztig’s constructible characters based on a Galois covering of the Calogero–Moser space). To compute the former, we devised an algorithm for determining generators of the center of the rational Cherednik algebra (this algorithm has several further applications), and to compute the latter we developed an algorithmic approach to the construction of cellular characters via Gaudin operators. We have implemented all our algorithms in the Cherednik Algebra Magma Package by the second author and used this to confirm open conjectures in several new cases. As an interesting application in birational geometry we are able to determine for many exceptional complex reflection groups the chamber decomposition of the movable cone of a $$\mathbb {Q}$$ Q -factorial terminalization (and thus the number of non-isomorphic relative minimal models) of the associated symplectic singularity. Making possible these computations was also a source of inspiration for the first author to propose several conjectures about the geometry of Calogero–Moser spaces (cohomology, fixed points, symplectic leaves), often in relation with the representation theory of finite reductive groups.

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19

Berest, Yuri, and Oleg Chalykh. "A∞-modules and Calogero-Moser spaces." Journal für die reine und angewandte Mathematik (Crelles Journal) 2007, no. 607 (January 27, 2007). http://dx.doi.org/10.1515/crelle.2007.046.

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20

Bonnafé, Cédric, and Ruslan Maksimau. "Fixed points in smooth Calogero–Moser spaces." Annales de l'Institut Fourier, June 7, 2021, 1–36. http://dx.doi.org/10.5802/aif.3404.

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21

Bonnafé, Cédric, and Peng Shan. "On the Cohomology of Calogero–Moser Spaces." International Mathematics Research Notices, March 28, 2018. http://dx.doi.org/10.1093/imrn/rny036.

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22

Voit, Michael. "Freezing limits for Calogero–Moser–Sutherland particle models." Studies in Applied Mathematics, August 4, 2023. http://dx.doi.org/10.1111/sapm.12628.

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AbstractOne‐dimensional interacting particle models of Calogero–Moser–Sutherland type with N particles can be regarded as diffusion processes on suitable subsets of like Weyl chambers and alcoves with second‐order differential operators as generators of the transition semigroups, where these operators are singular on the boundaries of the state spaces. The most relevant examples are multivariate Bessel processes and Heckman–Opdam processes in a compact and noncompact setting where in all cases, these processes are related to special functions associated with root systems. More precisely, the transition probabilities can be described with the aid of multivariate Bessel functions, Jack and Heckman–Opdam Jacobi polynomials, and Heckman–Opdam hypergeometric functions, respectively. These models, in particular, form dynamic eigenvalue evolutions of the classical random matrix models like β‐Hermite, β‐ Laguerre, and β‐Jacobi, that is, MANOVA, ensembles. In particular, Dyson's Brownian motions and multivariate Jacobi processes are included. In all cases, the processes depend on so‐called coupling parameters. We review several freezing limit theorems for these diffusions where, for fixed N, one or several of the coupling parameters tend to ∞. In many cases, the limits will be N‐dimensional normal distributions and, in the process case, Gauss processes. However, in some cases, normal distributions on half spaces and distributions related to some other ensembles appear as limits. In all cases, the limits are connected with the zeros of the classical one‐dimensional orthogonal polynomials of order N.

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23

Normatov, Zafar, and Rustam Turdibaev. "Calogero-Moser Spaces and the Invariants of Two Matrices of Degree 3." Transformation Groups, October 18, 2022. http://dx.doi.org/10.1007/s00031-022-09776-y.

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24

Berntson, Bjorn K., Ernest G. Kalnins, and Willard Miller. "Toward Classification of 2nd Order Superintegrable Systems in 3-Dimensional Conformally Flat Spaces with Functionally Linearly Dependent Symmetry Operators." Symmetry, Integrability and Geometry: Methods and Applications, December 16, 2020. http://dx.doi.org/10.3842/sigma.2020.135.

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We make significant progress toward the classification of 2nd order superintegrable systems on 3-dimensional conformally flat space that have functionally linearly dependent (FLD) symmetry generators, with special emphasis on complex Euclidean space. The symmetries for these systems are linearly dependent only when the coefficients are allowed to depend on the spatial coordinates. The Calogero-Moser system with 3 bodies on a line and 2-parameter rational potential is the best known example of an FLD superintegrable system. We work out the structure theory for these FLD systems on 3D conformally flat space and show, for example, that they always admit a 1st order symmetry. A partial classification of FLD systems on complex 3D Euclidean space is given. This is part of a project to classify all 3D 2nd order superintegrable systems on conformally flat spaces.

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