The Structure Of A Class Of Fixed-point Subalgebras Of Quantum Torus Lie Algebras

In this thesis,we investigate a class of fixed-point subalgebras L_qof the quantum torus Lie algebra.It is an infinite dimensional Lie algebra spanned by{L_m^α|α∈Z₊,m∈Z},subject to the following commutation relations[L_m^α,L_n^β]=[mβ-nα]_qL_m+n^α+β+（-1）^β[mβ+nα]_qL_m+n^α-β,?α,β∈Z₊,m,n∈Z.where L_m⁰=0,L_m^-α=-（-1）^αL_m^α.Suppose that q is not a root of unity,our main results can be stated as follows.（1）We classify all invariant symmetric bilinear forms on L_qand prove that Inv（L_q,C）=Cφ,whereφ（L_m^α,L_n^β）=-（-1）^αδ_α,βδ_m+n,0,?α,β∈Z₊,m,n∈Z.（2）We show that L_qis a finitely generated perfect Lie algebra and determine the firstcohomology group of the coefficients of L_qin the adjoint module.We proved thatH¹（L_q,L_q）=CD,whereD（L_m^α）=m L_m^α,?α∈Z₊,m∈Z.（3）We compute the second-order cohomology group with trivial coefficients for L_q:H²（L_q,C）=Cψ,whereψ（L_m^α,L_n^β）=-（-1）^αmδ_m+n,0δ_α-β,0,?α,β∈Z₊,m,n∈Z.We also determine the universal central extension of L_q.