A MATRIX REALIZATION OF THE QUANTUM GROUP 𝔤p,q

Abstract

In this paper we shall find a matrix realization of the quantum group 𝔤p,q introduced in [B. L. Feigin et al., Nucl. Phys. B757 (2006) 303–343]. For this purpose, we construct all primitive idempotents and a basis of 𝔤p,q. We determine the action of elements of the basis on indecomposable projective modules, that gives rise to a matrix realization of 𝔤p,q. By using this result, we obtain a basis of the space of symmetric linear functions on 𝔤p1,p2 and express the symmetric linear functions obtained by the left integral, the balancing element, and the center of 𝔤p,q in terms of this basis.