A quantum analog of the Z algebra
A quantum analog of the Z algebra
复制标题
Z 代数的量子模拟
DOI：
10.1063/1.531581
复制
发表时间：
1995
影响因子：
1.3
通讯作者：
L. Vinet
中科院分区：
文献类型：
--
作者：
A. Hamid Bougourzi;L. Vinet
我们定义了Z代数的一个自然量子类似物，我们称之为Zq代数，通过从具有水平k的量子仿射代数Uq（sl（2）■）中模出海森堡代数。讨论了Zq代数的表示理论。特别地，当k=1和k=2时，我们将其约化为群代数，以及群代数与量子Clifford代数的张量积，从而我们恢复了Frenkel-Jing和Bernard分别实现的Uq（sl（2）■）-标准模的显式构造。此外，对于任意非零水平k，我们证明了由Lepowsky和Primc构造的最简单Z-广义Verma模的显式基也是其对应的Zq-模的基，即，它在一般q的q变形下是不变的。我们期望这个Zq代数[在k层与Uq（sl（2）■）相关]在非临界Zk统计模型中扮演动力学对称的角色。
We define a natural quantum analog for the Z algebra, and which we refer to as the Zq algebra, by modding out the Heisenberg algebra from the quantum affine algebra Uq(sl(2)■) with level k. We discuss the representation theory of this Zq algebra. In particular, we exhibit its reduction to a group algebra, and to a tensor product of a group algebra with a quantum Clifford algebra when k=1, and k=2, and thus we recover the explicit constructions of Uq(sl(2)■)‐standard modules as achieved by Frenkel–Jing and Bernard, respectively. Moreover, for arbitrary nonzero level k, we show that the explicit basis for the simplest Z‐generalized Verma module as constructed by Lepowsky and Primc is also a basis for its corresponding Zq‐module, i.e., it is invariant under the q‐deformation for generic q. We expect this Zq algebra [associated with Uq(sl(2)■) at level k] to play the role of a dynamical symmetry in the off‐critical Zk statistical models.