Representations of Quantum Lorentz Group on Gelfand Spaces
Representations of Quantum Lorentz Group on Gelfand Spaces
复制标题
量子洛伦兹群在Gelfand空间上的表示
DOI：
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发表时间：
2000
期刊：
影响因子：
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通讯作者：
S. Woronowicz
中科院分区：
文献类型：
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作者：
W. Pusz;S. Woronowicz
找到并详细描述了量子Lorentz群QLG(允许岩泽分解的群)的一大类表示。在某种意义上，这个类包含了QLG的所有不可约酉表示。引入了QLG群的抛物子群P。它是由上三角矩阵组成的SL(2，C)的子群的光滑变形。给出了P的所有一维表示(特征标)的集合的刻划。证明了该集合的拓扑结构与经典Lorentz群的抛物线子群不同。研究了由抛物子群P的特征诱导的QLG的(一般非酉性)表示类。与经典情形一样，表示作用于齐次空间PQLG(Gelfand空间)上的(量子)线界的光滑部分的空间。对任意一对Gelfand空间，描述了所有非零不变双线性形式的集合。此集合不只对某些配对为空。我们给出了这样的配对的完整列表。利用这个列表，我们解决了表示的等价性和不可约性问题。我们区分了一类带有QLG的酉表示的Gelfand空间。
A large class of representations of the quantum Lorentz group QLG (the one admitting Iwasawa decomposition) is found and described in detail. In a sense the class contains all irreducible unitary representations of QLG. Parabolic subgroup P of the group QLG is introduced. It is a smooth deformation of the subgroup of SL (2, C) consisting of the upper-triangular matrices. A description of the set of all 1-dimensional representations (the characters) of P is given. It turns out that the topological structure of this set is not the same as for the parabolic subgroup of the classical Lorentz group. The class of (in general non-unitary) representations of QLG induced by characters of its parabolic subgroup P is investigated. Representations act on spaces of smooth sections of (quantum) line boundles over the homogeneous space PQLG (Gelfand spaces) as in the classical case. For any pair of Gelfand spaces the set of all non-zero invariant bilinear forms is described. This set is not empty only for certain pairs. We give a complete list of such pairs. Using this list we solve the problems of equivalence and irreducibility of the representation. We distinguish a class of Gelfand spaces carrying unitary representations of QLG.