On Semi-groups Arising from Quantum Groups

LetGqbe a quantum group. In this paper, we introduce and completely characterize a semi-group and its Borel sub-semi-groups consisting of algebra homomorphisms from the coordinate algebrak[Gq] ofGq. Also, we present invariant elements for the semi-group, which are closely connected to Hopf ideals ofk[Gq]. Finally, based on the theory of structures of the semi-group, we prove the existence of a Frobenius morphism overGqwhenever the parameterqis a root of unity.