A matrix realization of the quantum group g_{p, q}

In this paper we will find a matrix realizations of the quantum group g_{p, q}. For this purpose, we construct all primitive idempotents and a basis of g_{p, q}. We determine the action of elements of the basis on the indecomposable projective modules, which give rise to a matrix realization of g_{p, q}. By using this result, we obtain a basis of the space of symmetric linear functions on g_{p, q}} and express the symmetric linear functions obtained by the left integral, the balancing element and the center of g_{p, q} in term of this basis.