Simple Modules Of Quantum Groups For Type A₂

Let v be oneP^th priInitive root of unity withp an odd prime and we denote Z[v]by B, Q(v) by B'. We can define a quanum group for type A2 as a B' -algebra'5 which generated by E (i =l, 2, 3), F (i = l, 2, 3), K, (i = l, 2) under somerelations. In this paPer, we will give the structure for all 'U-modules (uP toisomorphism) and find out their B'-bases and dimensions. Furthermore, we alsoprove the Lusztigs conjectore (see reference [8]) for quantUIn grouPs, in case A2.