Representations Of Two Classes Of Vector Field Lie Algerbas

In this paper,we study irreducible weight modules with finite dimensional weight spaces over two classes of vector field Lie algebras.We firstly classify irreducible uniformly bounded weight modules for the vector field Lie algebra W∞ of infinite rank.It turns out that any such modules are intermediate series modules.This result is very different from the vector field Lie algebra Wd of finite rank.Next we study jet modules over the vector field Lie algebra g=Der(A)of finite rank,for A=C[t1,t1-1,t2].We show that there is an algebra isomorphism between the smash product A#U(g)and D(?)U(L),where D is the differential operator algebra C[t1,t1-1,t2,(?),(?)],and L is the jet Lie algebra of g.As an application,we give tensor product realizations of irreducible uniformly bounded jet modules over g.We show that any irreducible jet module is isomorphic to F(P,V).If P is an irreducible D-module and V is a finite dimensional irreducible gl2-module,then F(P,V)=P(?)C V is an irreducible jet module,but it is not always irreducible as a g-module.Finally we obtain the sufficient and necessary conditions for F(P,V)to be an irreducible module over g.