Massive Integrable Models and Infinite Dimensional Symmetries

为了回答“什么样的对称性保证了海量可积模型（量子场论和晶格模型）的可积性或无穷多守恒量”这一问题，我们一直在研究无限维对称性和可解晶格模型。与玻尔兹曼权值为Yang-Baxter方程椭圆解的可解晶格模型的面型和顶点型相对应，椭圆量子群具有面型和顶点型。它们的区别之一是能量特征值的计数；前者为均质级配，后者为主级配。一些面型模型得到了可解晶格模型的自由场实现，而顶点型模型没有直接实现。为了给这个问题提供一些提示，我们研究了具有主阶的（量子）仿射李代数。构造了具有主级配的sl^^^^_2的自由场实现。我们指出Lepowsky-Wilson的Z代数是变形Virasoro代数的一定极限，该代数是通过Cartan部分的分裂，在主阶上由sl^^^^_2得到更多的ch。换句话说，变形的Virasoro代数可以看作是Lepowsky-Wilson的Z代数的q变形。这个观察结果可以帮助我们研究顶点模型。对于高秩情况，通过显式计算变形W_N代数的定义关系，建立了sl_N版本的Z代数与变形W_N代数之间的相同关系。Calogero-Moser模型是经典力学和量子力学中非常有趣的可积模型，例如，它与Seiberg-Witten理论、超级Yang-Mills理论、变形Virasoro和W代数等有关。最近有人指出，即使在经典理论中，平衡位置上的各种量也被“量子化为整数值”。基于这一发现，我们计算了所有有限根系统具有有理和三角势的Calogero-Moser模型的经典平衡位置，并定义了Coxeter （Weyl）不变多项式。这些多项式的系数也是整数值。少

In order to answer the question "What symmetries ensure the integrability or infinitely many conserved quantities of massive integrable models (quantum field theory and lattice models)?", we have been studying infinite dimensional symmetries and solvable lattice models.Corresponding to the face type and the vertex type of solvable lattice models whose Boltzmann weights are elliptic solutions of the Yang-Baxter equation, elliptic quantum groups have the face type and the vertex type. One of their differences is the counting of the energy eigenvalues; the former is homogeneous gradation and the latter is principal gradation. Free filed realizations of solvable lattice models were obtained for some face type models but not for the vertex type models direct way. To get some hint for this problem, we studied (quantum) affine Lie algebra with principal gradation. We construct a free field realization of sl^^^^_2 with principal gradation. We point out that the Lepowsky-Wilson's Z algebra, whi … More ch is obtained from sl^^^^_2 with principal gradation by splitting the Cartan part, is certain limit of the deformed Virasoro algebra. In other word the deformed Virasoro algebra can be considered as a q-deformation of Lepowsky-Wilson's Z algebra. This observation may help us to study the vertex models. For higher rank case we establish the same relationship between the sl_N version of Z algebra and the deformed W_N algebra by calculating the defining relation of the deformed W_N algebra explicitly.Calogero-Moser models are very interesting integrable models as both classical and quantum mechanics, for example, they are related to the Seiberg-Witten theory of super Yang-Mills theory, the deformed Virasoro and W algebras, etc. Recently it is pointed out that various quantities at equilibrium positions are 'quantized in integer values' even in classical theory. Motivated by this observation, we calculate the classical equilibrium position of Calogero-Moser models with rational and trigonometric potential for all finite root systems, and define Coxeter (Weyl) invariant polynomials. Coefficients of these polynomials are also integer values. Less