The Structure Of A Class Of Fixed-point Subalgebras Of Virasoro-like Lie Algebras

In this thesis,we study a class of fixed-point subalgebra of the the Virasoro-like Lie algebra L.It is an infinite dimensional Lie algebra spanned by {L_m^?|??Z+,m?Z} with the Lie brackets:[L_m^?,L_n^?]=(m?-n?)L_m+n^?+?+(-1)^?(m?+n?)L_m+n^?-?,(?)?,??Z₊,m,n ?Z,where L_m⁰=0,L_m^-?=-(-1)?L_m^?.Our main results can be stated as follows:(1)We determine all symmetric invariant bilinear forms on L and prove that Inv(L)=C_?1?C_?2,where ?1,?2 is defined by,for (?)m,n? Z,?,??Z₊,(2)We compute all derivations of L and prove that H¹(L,L)=CD,where D is defined by D(L_m^?)=mL_m^?,(?)m?Z,??Z₊.(3)We obtain all one-dimensional central extensions of L and prove that H²(L,C)=C?,where ? is defined by:?(L_m^?,L_n^?)=-(-1)^?m?_m+n,0?_?,?,(?)m,n?Z,?,??Z₊.