论文标题
关于仰光的贝斯属bethians的经典限制
On classical limits of Bethe subalgebras in Yangians
论文作者
论文摘要
简单谎言代数$ \ mathfrak {g} $的yangian $ y（\ mathfrak {g}）$可以被视为两个不同的hopf代数的变形：通用的代数代数$ u（\ mathfrak {g mathfrak {g} [t]） $ \ MATHCAL {O}（g_1 [t^{ - 1}]]）$。这两个代数都是从扬吉亚人获得的，通过对扬吉的适当过滤进行相关的分级。伯特（Bethe）子代理构成了自然的换向子代理家庭，具体取决于伴随组$ g $的组元素$ c $。这些代数在基本表示的张量产物中的图像给出了量子XXX Heisenberg磁铁链的所有积分。我们将Bethe sibgebras的相关分级描述为$ U（\ Mathfrak {g} [t]）$中的子代理，以及$ \ Mathcal {o}（g_1 [g_1 [t^{ - 1}]）$ in G $ in G $。我们表明，分配给$ g $的身份的Bethe subalgebra的$ U（\ Mathfrak {g} [t]）$的关联是$ u（\ Mathfrak {g} [t]）$ u的通用Gaudin subalgebra。这将talalaev的公式概括为通用Gaudin subgerbra的发电机，以$ \ Mathfrak {g} $的任何类型的$。特别是，这表明可以对高丁磁铁链的较高的哈密顿人进行量化，而无需指临界水平的Feigin-Frenkel中心。利用我们对伯特级亚级别的相关等级的一般结果，我们计算了bethe subemalgebras的某些限制，与常规的semisimple $ c \ in G $相对应，因为$ c $转到了不规则的半密布组元素$ c_0 $。我们表明，此限制是较小的贝斯subgebra的产物，也是$ C_0 $的通用信封代数中参数subergebra的量子转移，$ c_0 $ in $ \ mathfrak {g} $。这概括了Vinberg在量化参数次代词的量化方面的Nazarov-Olshansky解决方案。
The Yangian $Y(\mathfrak{g})$ of a simple Lie algebra $\mathfrak{g}$ can be regarded as a deformation of two different Hopf algebras: the universal enveloping algebra $U(\mathfrak{g}[t])$ and the coordinate ring of the first congruence subgroup $\mathcal{O}(G_1[[t^{-1}]])$. Both of these algebras are obtained from the Yangian by taking the associated graded with respect to an appropriate filtration on Yangian. Bethe subalgebras 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 as subalgebras in $U(\mathfrak{g}[t])$ and in $\mathcal{O}(G_1[[t^{-1}]])$ for all semisimple $C\in G$. We show that associated graded in $U(\mathfrak{g}[t])$ of the Bethe subalgebra assigned to the identity of $G$ is the universal Gaudin subalgebra of $U(\mathfrak{g}[t])$ obtained from the center of the corresponding affine Kac-Moody algebra at the critical level. This generalizes Talalaev's formula for generators of the universal Gaudin subalgebra to $\mathfrak{g}$ 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 associated graded of Bethe subalgebras, we compute some limits of Bethe subalgebras corresponding to regular semisimple $C\in G$ as $C$ goes to an irregular semisimple group element $C_0$. We show that this limit is the product of the smaller Bethe subalgebra and a quantum shift of argument subalgebra in the universal enveloping algebra of the centralizer of $C_0$ in $\mathfrak{g}$. This generalizes the Nazarov-Olshansky solution of Vinberg's problem on quantization of shift of argument subalgebras.
