In recent years, one of the new applications of the coherent state method was to construct representation of superalgebras and quantum superalgebras. Following this stream, we had a contribution to working out explicit representation of Uq[gl(2|1)]. Up to now, Uq[gl(2|1) is still the biggest quantum superalgebra representations in coherent state basis of which can be built. In this article, we will show some detailed techniques used in our previous work but useful for our further investigations. | Communications in Physics, Vol. 23, No. 1 (2013), pp. 29-37 ON THE COHERENT STATE METHOD IN CONSTRUCTING REPRESENTATIONS OF QUANTUM SUPERALGEBRAS NGUYEN CONG KIEN Institute of Physics, VAST NGUYEN ANH KY Institute of Physics, VAST and Laboratory for High Energy Physics and Cosmology, Faculty of Physics, College of Science, Vietnam National University, Hanoi Abstract. In recent years, one of the new applications of the coherent state method was to construct representation of superalgebras and quantum superalgebras. Following this stream, we had a contribution to working out explicit representation of Uq [gl(2|1)]. Up to now, Uq [gl(2|1) is still the biggest quantum superalgebra representations in coherent state basis of which can be built. In this article, we will show some detailed techniques used in our previous work but useful for our further investigations. The newest results on building representations in a coherent state basis of Uq [osp(2|2)], which has the same rank as Uq [gl(2|1)], are also briefly exposed. I. INTRODUCTION In the late 1920’s, the concept of coherent states (CS’s) was introduced by E. Schr¨odinger [1] while searching for a classical analog of quantum states of quantum harmonic oscillators. For more than 80 years, the concept of CS’s has been developed by many people, especially, a crutial step was made by A. Perelomov who generalized the CS concept for arbitrary Lie algebras [2–4]. This concept was also extended to that of vector coherent states (VCS’s) [5–9]. In 1970’s, the formation of supersymmetry (SUSY) led to the creation of a new research trend in physics and mathematics (although, presently, the SUSY phenomenological models are in a difficult time when the latest results of the LHC have not been able to confirm them). Combining with the SUSY idea, the concept of CS’s was developed to those of super coherent states (SCS’s) and supervector coherent states (SVCS’s) [10–14]. With special characteristics, CS’s are quantum states having
