Representation Theory For Cartan Type W Lie (Super) Algebras

Let Am,n(resp.Am,n+)be the Laurent polynomial superalgebra(resp.polynomial superalgebra)generated by m even elements and n odd elements,and let Wm,n(resp.Wm,n+)be the Lie superalgebra of superderivations of Am,n(resp.Am,n+).This paper studies the representation theory of the Witt superalgebra Wm,n and Wm,n+,which includes:(1)The irreduciblity of the tensor module F(P,M),where P is a simple module over the Weyl superalgebra,M is a simple weight module over the general linear Lie superalgebra gl(m,n).We determine the sufficient and necessary conditions for F(P,M)to be simple.Moreover,we give a composition series of F(P,M).(2)The classification of simple strong Harish-Chandra modules over Wm,n.We prove that any such module is isomorphic to a simple quotient of a tensor module,or isomorphic to a simple weight module of highest weight type that is induced by a tensor module.(3)The classification of simple bounded modules over Wm,n+.We prove that any such module is isomorphic to a simple quotient of a tensor module F(P,L(V1(?)V2)),where P is a simple weight module over the Weyl superalgebra K is a finite dimensional simple glm-module,V2 is a simple bounded gln-module,and L(V1(?)V2)is the unique simple quotient of the Kac module of V1(?)V2.(4)The classification of simple strong Harish-Chandra modules over Wm,n+.We prove that any such module is isomorphic to the unique simple submodule of a strong Harish-Chandra tensor module.