We　classify　all　the　Hopf　PBW-deformations　of　a　new　type　quantum　group　U-q(sl(2)*)　from　which　the　classical　Drinfeld-Jimbo　quantum　group　U-q(sl(2)*)　can　arise　as　an　almost　unique　nontrivial　one.　Different　from　the　U-q(sl(2)*)　case,　the　category　of　finite-dimensional　U-q(sl(2)*)-modules　is　non-semisimple.　We　establish　a　block　decomposition　theorem　for　the　category　U-q(sl(2)*)-mod(wt)　of　finite-dimensional　weight　modules　of　U-q(sl(2)*).　On　the　level　of　tensor　category,　we　show　that　U-q(sl(2)*)-mod(wt)　(respectively,　the　category　U-q(sl(2))-mod　of　finite-dimensional　U-q(sl(2)*))-modules)　can　be　realized　via　(respectively,　deformed)　preprojective　algebras　of　Dynkin　type　A.