Of Uq (of Sl <sub> 2 </ Sub>) I Road, Hopf Algebra,

In this dissertation, a quiver D of the Hopf algebra Uq(sl₂), when q is not a root of unity, which is a one-parameter deformation of the enveloping algebra of the lie algebra sl₂ is constructed. And the comultiplication of the paths in the vector space kD is defined. It is proved that the comultiplication of the paths is well-defined. So the path coalgebra structure on the vector space kD is obtained, and it is denoted by kD^C in this dissertation. Morever, a multiplication of paths on the path coalgebra kD^C is defined, and it is proved that this multiplication is well-defined, satisfies the associative law of multiplication, furthermore it is a coalgebra morphism of co-tensor product kD^C (?) kD^C to path coalgebra kD^C. Therefore the Hopf algebra structure of path coalgebra kD^C is obtained, it is called Co-path Hopf algebra of Uq(sl₂). And in section four, several graded Hopf algebra structure is listed, compares with the Hopf algebra structure of Uq(sl₂), the conclution that Uq(sl₂), as a Hopf algebra can't embed in its Co-path Hopf algebrak.D^C, even though it can embed in its path coalgebra as a coalgebra is obtained, which explains the difference of the structure of Hopf algebra between Uq(sl₂) and its Co-path Hopf algebrak D^C.Furthermore, a graded quotient Hopf algebra is found, gr(Uq(sl₂))} and it is proved that it can embed in the copath Hopf algebra of Uq(sl₂), and lists a base of gr(Uq(sl₂))t which is the conclution obtained in this dissertation.