论文标题
Baxter Q功能和融合标志的扩展系统I：简单的情况
Extended systems of Baxter Q-functions and fused flags I: simply-laced case
论文作者
论文摘要
可集成模型的光谱通常是根据满足功能关系的光谱参数的通勤函数来编码的。我们建议通过扩展的Q系统以协变量的方式来描述这种交换的代数，该Q系统包含Q-矢量的每个（Langlands dual）基础对称代数的（Langlands dual）的每个基本表示。这些Q向量是参数化的一个完整标志的集合，这些标志以特定方式互相融合。我们表明，融合标志是在特定规格中运行的有限差分，明确的识别取决于Coxeter元素的选择。 本文考虑了简单的代数的情况，并带有简单的Dynkin图。对于$ a_r $系列，构建与文献中已经知道的结果相吻合。我们将拟议的形式主义应用于$ d_r $系列和特殊代数$ e_r $，$ r = 6,7,8 $的情况。特别是，我们根据Q-功能求解了双线性方程，并在$ d_r $ case中给出了扩展Q系统的明确字符解决方案。我们还展示了如何通过类似于$ a_r $方案的过程来构建$ d_r $类型的扩展Q-System，但是该过程构建了各向同性空间的融合标志，或者是通过Fused Fierz关系构建的。 最后，对于理性，三角和椭圆旋转链的情况，我们为扩展Q-System的Q-功能的分析结构提出了一个明确的ANSATZ。我们猜想以这种方式约束的扩展Q系统始终与这些模型的通勤传输矩阵的伯特代数进行培养，此外，可以用来证明Bethe代数具有简单的关节频谱。
The spectrum of integrable models is often encoded in terms of commuting functions of a spectral parameter that satisfy functional relations. We propose to describe this commutative algebra in a covariant way by means of the extended Q-system that comprise Q-vectors in each of the fundamental representations of the (Langlands dual of) the underlying symmetry algebra. These Q-vectors turn out to parameterise a collection of complete flags which are fused with one another in a particular way. We show that the fused flag is a finite-difference oper in a particular gauge, explicit identification depends on a choice of a Coxeter element. The paper considers the case of simple Lie algebras with a simply-laced Dynkin diagram. For the $A_r$ series, the construction coincides with already known results in the literature. We apply the proposed formalism to the case of the $D_r$ series and the exceptional algebras $E_r$, $r=6,7,8$. In particular, we solve Hirota bilinear equations in terms of Q-functions and give the explicit character solution of the extended Q-system in the $D_r$ case. We also show how to build up the extended Q-system of $D_r$ type starting either from vectors, by a procedure similar to the $A_r$ scenario which however constructs a fused flag of isotropic spaces, or from pure spinors, \via fused Fierz relations. Finally, for the case of rational, trigonometric, and elliptic spin chains, we propose an explicit ansatz for the analytic structure of Q-functions of the extended Q-system. We conjecture that the extended Q-system constrained in such a way is always in bijection with the Bethe algebra of commuting transfer matrices of these models and moreover can be used to show that the Bethe algebra has a simple joint spectrum.
