arXiv math-ph/05090322005-09-15math-phmath.MP

q-变形量子李代数

q-Deformed quantum Lie algebras

Alexander Schmidt, Hartmut Wachter

备注：: Latex, 49 pages, no figures
DOI：: 10.1016/j.geomphys.2005.12.003

摘要：

重点研究了具有物理意义的q-变形量子代数，即$U_{q}(su_{2})$，$U_{q}(so_{4})$和q-变形Lorentz代数.这篇文章的主要关注点是组装这些对称代数的重要思想在一个一致的框架，应作为物理学，特别是量子场论的表示论研究的出发点。在每一种情况下，考虑从实现对称发电机内的微分代数。列出了对称生成元上的余积和对映体的计算公式。对称性生成元的作用在它们的Hopf结构中被看作是经典的对称性生成元的q-模拟，并被明确地写出来。计算了对称生成元的旋量和矢量表示。本文综述了对称生成元与旋量或矢量算符分量之间的对易关系。相应的量子李代数的关系计算。它们的Casimir算子以类似于未变形情况的形式写出。

英文摘要：

Attention is focused on q-deformed quantum algebras with physical importance, i.e. $U_{q}(su_{2})$, $U_{q}(so_{4})$ and q-deformed Lorentz algebra. The main concern of this article is to assemble important ideas about these symmetry algebras in a consistent framework which shall serve as starting point for representation theoretic investigations in physics, especially quantum field theory. In each case considerations start from a realization of symmetry generators within the differential algebra. Formulae for coproducts and antipodes on symmetry generators are listed. The action of symmetry generators in terms of their Hopf structure is taken as q-analog of classical commutators and written out explicitly. Spinor and vector representations of symmetry generators are calculated. A review of the commutation relations between symmetry generators and components of a spinor or vector operator is given. Relations for the corresponding quantum Lie algebras are computed. Their Casimir operators are written down in a form similar to the undeformed case.

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