Representations of quantum superalgebra Uq[gl(2|1)] in a coherent state basis and generalization

The coherent state method has proved to be useful in quantum physics and mathematics. This method, more precisely, the vector coherent state method, has been used by some authors to construct representations of superalgebras but almost, to our knowledge, it has not yet been extended to quantum superalgebras, except \(U_q[osp(1|2)]\), one of the smallest quantum superalgebras. In this article the method is applied to a bigger quantum superalgebra, namely \(U_q[gl(2|1)]\), in constructing \(q\)--boson-fermion realizations and finite-dimensional representations which, when irreducible, are classified into typical and nontypical representations. This construction leads to a more general class of \(q\)--boson-fermion realizations and finite-dimensional representations of \(U_q[gl(2|1)]\) and, thus, at \(q=1\), of \(gl(2|1)\). Both \(gl(2|1)\) and \(U_q[gl(2|1)]\) have found different physics applications, therefore, it is meaningful to construct their representations.