The Yangian Y (𝔤) of a simple Lie algebra 𝔤 can be regarded as a deformation of two different Hopf algebras: the universal enveloping algebra of the current algebra \( U\left(\mathfrak{g}\left[t\right]\right) \) and the coordinate ring of the first congruence subgroup \( \mathcal{O}\left({G}_1\left[\left[{t}^{-1}\right]\right]\right). \) Both of these algebras are obtained from the Yangian by taking the associated graded with respect to an appropriate filtration on Y (𝔤).

Bethe subalgebras B(C) in Y (𝔤) form a natural family of commutative subalgebras depending on a group element C of the adjoint group G. The images of these algebras in tensor products of fundamental representations give all integrals of the quantum XXX Heisenberg magnet chain.

We describe the associated graded of Bethe subalgebras in the Yangian Y (𝔤) of a simple Lie algebra 𝔤 as subalgebras in \( U\left(\mathfrak{g}\left[t\right]\right) \) and in \( \mathcal{O}\left({G}_1\left[\left[{t}^{-1}\right]\right]\right) \) for all semisimple C ∈ G. In particular, we show that the associated graded in \( U\left(\mathfrak{g}\left[t\right]\right) \) of the Bethe subalgebra B(E) assigned to the unity element of G is the universal Gaudin subalgebra of \( U\left(\mathfrak{g}\left[t\right]\right) \) obtained from the center of the corresponding affine Kac–Moody algebra \( \hat{\mathfrak{g}} \) at the critical level. This generalizes Talalaev’s formula for generators of the universal Gaudin subalgebra to 𝔤 of any type. In particular, this shows that higher Hamiltonians of the Gaudin magnet chain can be quantized without referring to the Feigin–Frenkel center at the critical level.

Using our general result on the associated graded of Bethe subalgebras, we compute some limits of Bethe subalgebras corresponding to regular semisimple C ∈ G as C goes to an irregular semisimple group element C₀. We show that this limit is the product of the smaller Bethe subalgebra B(C₀) and a quantum shift of argument subalgebra in the universal enveloping algebra of the centralizer of C₀ in 𝔤. This generalizes the Nazarov–Olshansky solution of Vinberg’s problem on quantization of the (Mishchenko–Fomenko) shift of argument subalgebras.

简单李代数Yang的Yangian Y（𝔤）可以看作是两个不同的Hopf代数的变形：当前代数\（U \ left（\ mathfrak {g} \ left [t \ right] \ right）\）和第一个同等子组\（\ mathcal {O} \ left（{G} _1 \ left [\ left [{t} ^ {-1} \ right] \ right] \ right的坐标环）\）这两个代数均是通过对Y（𝔤）进行适当过滤而获得的相关等级从Yangian获得的。

贝特代数乙（Ç）在Ý（𝔤）形成取决于一组元素交换子代数的自然家庭ç伴随组G ^。这些代数在基本表示的张量积中的图像给出了量子XXX海森堡磁链的所有积分。

我们将简单Lie代数Yang的Yangian Y（𝔤）中的Bethe子代数的相关等级描述为\（U \ left（\ mathfrak {g} \ left [t \ right] \ right）\）和\中的子代数（\ mathcal {ö} \左（{G} _1 \左[\左[{吨} ^ { - 1} \右] \右] \右）\）对于所有半单ç ∈ ģ。特别是，我们证明了分配给G的单位元素的Bethe子代数B（E）的\（U \ left（\ mathfrak {g} \ left [t \ right] \ right）\）中的相关等级是\（U \ left（\ mathfrak {g} \ left [t \ right] \ right）\）的通用Gaudin子代数从临界级的相应仿射Kac-Moody代数\（\ hat {\ mathfrak {g}} \）的中心获得。这将通用高丁子代数的生成器的Talalaev公式推广为任何类型的𝔤。特别是，这表明可以在不参考临界水平的Feigin-Frenkel中心的情况下对高丁磁链的更高的哈密顿量进行量化。

在相关联的分级贝特子代数的使用我们的一般的结果，我们计算对应于常规半单贝特子代数的一些限制Ç ∈ ģ作为Ç变为不规则的半单组元素Ç ₀。我们证明了这个极限是较小的Bethe子代数B（C ₀）和₀中的C ₀的中心包围物的通用包络代数中自变量子代数的量子移位的乘积。这概括了范伯格问题的Nazarov–Olshansky解关于量化自变量子代数（Mishchenko–Fomenko）位移的问题。