Associated With Von Norman Generate Sub-generalized Quantum Envelope Algebra

This dissertation consists of five sections. We shall investigate extended quantum enveloping algebras with Von Neumann regular generators.The first and second sections are introduction and preliminaries respectively.In the third section,to begin with,from the algebra Uq(φ(k,h))[25],φ(k,h)is a polynomial,we introduce four von Neumann regular generators K,K,L,L.DefineP= KK,Q=LL)then they will be idempotents and obtain the corresponding Pierce decom-position.Two algebrasUK,L,norm(alg)(φ(K+L,H))and UK,L,twist(φ(K+L,H))will be got through the morphismΦ(37).Then we have the Pierce decomposition of UK,L,norm(alg)(φ(K+ L,H))andUK,L,twist(φ(K+L,H))and the Poincare-Birkhoff-Witt basis.In the fourth section,we investigate their coalgebra structures and give the condition which makes them be bialgebras.Next we give two appropriate antipodes TandS,then obtain two algebrasUK,L(VN-Hopf)(φ(K+L,H))and UK,L(Hopf)(φ(K+L,H)),the former is von Neumann Hopf algebra,the latter is Hopf algebra.In the fifth section,we investigate the center Z(Uq(φ(K+L,H)))ofUq(φ(K+L,H)) which including UK,L(Hopf)(φ(K+L,H))andUK,L(VN-Hopf)(φ(K+L,H))and give the format of Z(Uq(φ(K+L,H))).