Abstract
In this thesis, we prove that in the embedding of the Drinfeld double Dsl_n in the quantum torus algebra D_g, there are certain subsets of vertices such that the embedding of the rescaled Chevalley generators e_i of the Drinfeld double remains polynomial under mutations at those vertices. We also describe a quantum analogue of the Lusztig coordinates for different reduced words of the longest element when the semisimple Lie algebra is of type A₃. Using this we show that the embedding of the Drinfeld double is independent of the choice of reduced word of the longest element for types A, D, and E.