{"title":"三角Gaudin算子和动力学Hamiltonian算子的差分算子和对偶性","authors":"F. Uvarov","doi":"10.3842/SIGMA.2022.081","DOIUrl":null,"url":null,"abstract":"We study the difference analog of the quotient differential operator from [Tarasov V., Uvarov F., Lett. Math. Phys. 110 (2020), 3375-3400, arXiv:1907.02117]. Starting with a space of quasi-exponentials $W=\\langle \\alpha_{i}^{x}p_{ij}(x),\\, i=1,\\dots, n,\\, j=1,\\dots, n_{i}\\rangle$, where $\\alpha_{i}\\in{\\mathbb C}^{*}$ and $p_{ij}(x)$ are polynomials, we consider the formal conjugate $\\check{S}^{\\dagger}_{W}$ of the quotient difference operator $\\check{S}_{W}$ satisfying $\\widehat{S} =\\check{S}_{W}S_{W}$. Here, $S_{W}$ is a linear difference operator of order $\\dim W$ annihilating $W$, and $\\widehat{S}$ is a linear difference operator with constant coefficients depending on $\\alpha_{i}$ and $\\deg p_{ij}(x)$ only. We construct a space of quasi-exponentials of dimension $\\operatorname{ord} \\check{S}^{\\dagger}_{W}$, which is annihilated by $\\check{S}^{\\dagger}_{W}$ and describe its basis and discrete exponents. We also consider a similar construction for differential operators associated with spaces of quasi-polynomials, which are linear combinations of functions of the form $x^{z}q(x)$, where $z\\in\\mathbb C$ and $q(x)$ is a polynomial. Combining our results with the results on the bispectral duality obtained in [Mukhin E., Tarasov V., Varchenko A., Adv. Math. 218 (2008), 216-265, arXiv:math.QA/0605172], we relate the construction of the quotient difference operator to the $(\\mathfrak{gl}_{k},\\mathfrak{gl}_{n})$-duality of the trigonometric Gaudin Hamiltonians and trigonometric dynamical Hamiltonians acting on the space of polynomials in $kn$ anticommuting variables.","PeriodicalId":49453,"journal":{"name":"Symmetry Integrability and Geometry-Methods and Applications","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2022-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"Difference Operators and Duality for Trigonometric Gaudin and Dynamical Hamiltonians\",\"authors\":\"F. Uvarov\",\"doi\":\"10.3842/SIGMA.2022.081\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We study the difference analog of the quotient differential operator from [Tarasov V., Uvarov F., Lett. Math. Phys. 110 (2020), 3375-3400, arXiv:1907.02117]. Starting with a space of quasi-exponentials $W=\\\\langle \\\\alpha_{i}^{x}p_{ij}(x),\\\\, i=1,\\\\dots, n,\\\\, j=1,\\\\dots, n_{i}\\\\rangle$, where $\\\\alpha_{i}\\\\in{\\\\mathbb C}^{*}$ and $p_{ij}(x)$ are polynomials, we consider the formal conjugate $\\\\check{S}^{\\\\dagger}_{W}$ of the quotient difference operator $\\\\check{S}_{W}$ satisfying $\\\\widehat{S} =\\\\check{S}_{W}S_{W}$. Here, $S_{W}$ is a linear difference operator of order $\\\\dim W$ annihilating $W$, and $\\\\widehat{S}$ is a linear difference operator with constant coefficients depending on $\\\\alpha_{i}$ and $\\\\deg p_{ij}(x)$ only. We construct a space of quasi-exponentials of dimension $\\\\operatorname{ord} \\\\check{S}^{\\\\dagger}_{W}$, which is annihilated by $\\\\check{S}^{\\\\dagger}_{W}$ and describe its basis and discrete exponents. We also consider a similar construction for differential operators associated with spaces of quasi-polynomials, which are linear combinations of functions of the form $x^{z}q(x)$, where $z\\\\in\\\\mathbb C$ and $q(x)$ is a polynomial. Combining our results with the results on the bispectral duality obtained in [Mukhin E., Tarasov V., Varchenko A., Adv. Math. 218 (2008), 216-265, arXiv:math.QA/0605172], we relate the construction of the quotient difference operator to the $(\\\\mathfrak{gl}_{k},\\\\mathfrak{gl}_{n})$-duality of the trigonometric Gaudin Hamiltonians and trigonometric dynamical Hamiltonians acting on the space of polynomials in $kn$ anticommuting variables.\",\"PeriodicalId\":49453,\"journal\":{\"name\":\"Symmetry Integrability and Geometry-Methods and Applications\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2022-02-13\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Symmetry Integrability and Geometry-Methods and Applications\",\"FirstCategoryId\":\"101\",\"ListUrlMain\":\"https://doi.org/10.3842/SIGMA.2022.081\",\"RegionNum\":3,\"RegionCategory\":\"物理与天体物理\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Symmetry Integrability and Geometry-Methods and Applications","FirstCategoryId":"101","ListUrlMain":"https://doi.org/10.3842/SIGMA.2022.081","RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
Difference Operators and Duality for Trigonometric Gaudin and Dynamical Hamiltonians
We study the difference analog of the quotient differential operator from [Tarasov V., Uvarov F., Lett. Math. Phys. 110 (2020), 3375-3400, arXiv:1907.02117]. Starting with a space of quasi-exponentials $W=\langle \alpha_{i}^{x}p_{ij}(x),\, i=1,\dots, n,\, j=1,\dots, n_{i}\rangle$, where $\alpha_{i}\in{\mathbb C}^{*}$ and $p_{ij}(x)$ are polynomials, we consider the formal conjugate $\check{S}^{\dagger}_{W}$ of the quotient difference operator $\check{S}_{W}$ satisfying $\widehat{S} =\check{S}_{W}S_{W}$. Here, $S_{W}$ is a linear difference operator of order $\dim W$ annihilating $W$, and $\widehat{S}$ is a linear difference operator with constant coefficients depending on $\alpha_{i}$ and $\deg p_{ij}(x)$ only. We construct a space of quasi-exponentials of dimension $\operatorname{ord} \check{S}^{\dagger}_{W}$, which is annihilated by $\check{S}^{\dagger}_{W}$ and describe its basis and discrete exponents. We also consider a similar construction for differential operators associated with spaces of quasi-polynomials, which are linear combinations of functions of the form $x^{z}q(x)$, where $z\in\mathbb C$ and $q(x)$ is a polynomial. Combining our results with the results on the bispectral duality obtained in [Mukhin E., Tarasov V., Varchenko A., Adv. Math. 218 (2008), 216-265, arXiv:math.QA/0605172], we relate the construction of the quotient difference operator to the $(\mathfrak{gl}_{k},\mathfrak{gl}_{n})$-duality of the trigonometric Gaudin Hamiltonians and trigonometric dynamical Hamiltonians acting on the space of polynomials in $kn$ anticommuting variables.
期刊介绍:
Scope
Geometrical methods in mathematical physics
Lie theory and differential equations
Classical and quantum integrable systems
Algebraic methods in dynamical systems and chaos
Exactly and quasi-exactly solvable models
Lie groups and algebras, representation theory
Orthogonal polynomials and special functions
Integrable probability and stochastic processes
Quantum algebras, quantum groups and their representations
Symplectic, Poisson and noncommutative geometry
Algebraic geometry and its applications
Quantum field theories and string/gauge theories
Statistical physics and condensed matter physics
Quantum gravity and cosmology.