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^α|α∈Z+,m∈Z} with the Lie brackets:[L_m^α,L_n^β]=（mβ-nα）L_m+n^α+β+（-1）^β（mβ+nα）L_m+n^α-β,（?）α,β∈Z₊,m,n ∈Z,where L_m⁰=0,L_m^-α=-（-1）αL_m^α.Our main results can be stated as follows:（1）We determine all symmetric invariant bilinear forms on L and prove that Inv（L）=C_φ1⊕C_φ2,where φ1,φ2 is defined by,for （?）m,n∈ Z,α,β∈Z₊,（2）We compute all derivations of L and prove that H¹（L,L）=CD,where D is defined by D（L_m^α）=mL_m^α,（?）m∈Z,α∈Z₊.（3）We obtain all one-dimensional central extensions of L and prove that H²（L,C）=Cφ,where φ is defined by:φ（L_m^α,L_n^β）=-（-1）^αmδ_m+n,0δ_α,β,（?）m,n∈Z,α,β∈Z₊.