论文标题
差异运算符和三角gaudin和动态性哈密顿的二元性
Difference Operators and Duality for Trigonometric Gaudin and Dynamical Hamiltonians
论文作者
论文摘要
我们研究了[Tarasov V.，Uvarov F.，Lett的商差分运算符的差异类似物。数学。物理。 110（2020），3375-3400，Arxiv：1907.02117]。从Quasi Exponentials的空间开始，$ W = \langleα_{i}^{X} p_ {ij}（ij}（x），\，\，i = 1，\ dots，n，n，\，j = 1，j = 1，\ dots，n_} $ p_ {ij}（x）$是多项式，我们考虑正式的共轭$ \ check {s}^{\ dagger} _ {w} _ {w} $的差异差异操作员$ \ check {s} s} _ {w} _ {w} $ actallying $ \\ saffectying $ \ \ \ widehat {s}在这里，$ s_ {w} $是订单$ \ dim w $ nihihilating $ w $的线性差异操作员，而$ \ widehat {s} $是一个线性差异操作员，其恒定系数取决于$α__{i} $和$°p_ p_ p_ {ij}（x）$。我们构造了尺寸的准表达$ \ operatoRatorname {orparatorname {orpataTorname {orpect} \ check {s}^{\ dagger} _ {w} $，它由$ \ check {s}}^{\ dagger {\ dagger} _ {w} _ {我们还考虑了与准多项式空间相关的差分运算符的类似结构，这是表格$ x^{z} q（x）$的函数的线性组合，其中$ z \ in \ mathbb c $和$ q（x）$是多项式的。将我们的结果与[Mukhin E.，Tarasov V.，Varchenko A.，Adv。数学。 218（2008），216-265，arxiv：Math.qa/0605172]，我们将商的差异操作员的构造与$（\ Mathfrak {gl} _ {k} _ {k}，\ mathfrak {gl Mathfrak {gl} _ {n}）$ - {n}）$ - hamilton trigud hamilton and trigud hamilton and trigud hamilton and trigud hamilton和多项式的空间$ kn $抗议变量。
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 $W=\langle α_{i}^{x}p_{ij}(x),\, i=1,\dots, n,\, j=1,\dots, n_{i}\rangle$, where $α_{i}\in{\mathbb C}^{*}$ 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 $α_{i}$ and $°p_{ij}(x)$ 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.
