Hopf代数量子链及Jones型C*-基本构造

In this project, we want to study some problems which are closely related with the internal symmetry of the quantum field algebra and the corresponding Jones type C*-basic construction in Hopf algebra quantum chains (i.e. Hopf spin models). On the one hand, we will study the internal symmetry of the field algebra in Hopf spin models. We first construct the field algebra in Hopf spin models, and then define a left action of the Drinfeld double D(H) of the Hopf C*-algebra H on the field algebra. Moreover, we show that the observable algebra by the observational data is a nonzero C*-subalgebra of the field algebra, which just corresponds to one dimensional irreducible representation of the Drinfeld double D(H). At last, we discuss the internal symmetry of the field algebra in Hopf spin models using the representation theory. And on the other, we will study the Watatani index and Jones type C*-basic construction in Hopf spin models. We first construct the quasi-basis for the conditional expectation from the field algebra onto the observable subalgebra by the invariant Haar integral in D(H), from which the Watatani index for this conditional expectation can be given. Then we also consider the concrete form of the Jones type C*-basic construction for the field algebra and the observable subalgebra, and get the Jones tower in Hopf spin models.

本项目拟围绕Hopf代数量子链(即Hopf旋模型)中量子场代数的内部对称性以及相应的Jones型C*-基本构造进行研究。一方面，研究Hopf旋模型中场代数的内部对称性：首先，构造Hopf旋模型中的场代数，并定义Hopf C*-代数H的Drinfeld量子偶D(H)在场代数上的左作用；其次，证明由观测量数据构成的观测量代数是场代数的非零C*-子代数，并且它正好对应Drinfeld量子偶D(H)的一维不可约表示；最后，以表示论为工具讨论Hopf旋模型中场代数的内部对称性。另一方面，研究Hopf旋模型中的Watatani指标和Jones型C*-基本构造：利用D(H)中的不变Haar积分构造从场代数到观测量子代数的条件期望的拟基，由此得到该条件期望的Watatani指标；进而考虑场代数和观测量子代数的Jones型C*-基本构造的具体形式，并得到Hopf旋模型中的Jones塔。

研究工作主要包含：.1、研究有限群G的Drinfeld量子偶D(G)及由子群所确定的Hopf *-子代数D(G;H)的Jones型基本构造，并给出其具体形式。.2、通过定义Hopf C*-代数H的Drinfeld量子偶D(H)在Hopf旋模型场代数上的作用，得到交叉积C*-代数，最后证明该C*-代数就是Hopf旋模型中观测量代数和场代数对的Jones型基本构造，从而说明此基本构造不依赖于场代数的选取。.3、首先定义了由Hopf *-子代数所确定的Hopf旋模型，进而给出局部观测量代数上的右D(H,H1)-余模代数作用，得到由Hopf *-子代数所确定的Hopf旋模型的场代数。最后证明场代数是D(H,H1)-模代数，并且给出由Hopf *-子代数所确定的Hopf旋模型内部对称性，即存在D(H,H1)的唯一的C*-表示使得D(H,H1)和观测量代数互为换位。此结论推广了G-旋模型中的对偶定理。.此外，还研究冯•诺依曼代数的可测算子的性质以及量子相干的运算性质。