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^?|??Z₊,m?Z},subject to the following commutation relations[L_m^?,L_n^?]=[m?-n?]_qL_m+n^?+?+(-1)^?[m?+n?]_qL_m+n^?-?,??,??Z₊,m,n?Z.where L_m⁰=0,L_m^-?=-(-1)^?L_m^?.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)=C?,where?(L_m^?,L_n^?)=-(-1)^??_?,??_m+n,0,??,??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^?)=m L_m^?,???Z₊,m?Z.(3)We compute the second-order cohomology group with trivial coefficients for L_q:H²(L_q,C)=C?,where?(L_m^?,L_n^?)=-(-1)^?m?_m+n,0?_?-?,0,??,??Z₊,m,n?Z.We also determine the universal central extension of L_q.