Quantum Torus High Rank Of The Virasoro-like Algebra

It is well known that affine Kac—Moody algebra and their representations play important roles in many branches in both mathematics and theoretical physics. We notice that the quantum torus contains the Laurent polynomial ring as its special case, in the meanwhile the derivation Lie algebra of the quantum tours also contains some particular subalgebras. The paper aims at studying the derivation Lie algebra of a certain subalgebra of the derivation Lie algebra of the quantum tours.We have choosen a certain integeral d≥2. The vector space (?) owns e1, e2…ed as its base in the field of complex number C. q is choosen to be a certain nonzero complex number and qp equals to 1. Cq is a corresponding quantum tour which is associative but non-commutative algebra on C. LetΓ=∑i=1~d Zei.σand f are functions fromΓ×Γto C which satisfy the equationσ(n, m)=Π1≤i≤j≤dqnjmi and f(n, m)=σ(n, m)σ(m, n)-1.Define degree derivation deri satisfying deri(xn)= njxn, where xn=x1n1 x2n2…xdnd for n=∑i=1dniei∈Γ. Denote rad(f)={n∈Γ:f(n, m)= 1,(?)Γ} the radical of f. Let D (u, n)=xn∑i=1duideri and g=〈D(u, r):(u, r)= 0,0≠τ∈rad(f)〉. Easy to prove that g is the subalgebra of Der(Cq). We call it Virasoro - like algebra on quantum tours.Denote Derg the derivation Lie algebra of g. The paper is to characterize its form by using the results of Der(Cq). The paper is composed of three parts. First of all, in the chapter one we introduce the problem's current situation, its aim and mean. In the following, we give the definition of Virasoro - like algebra and some useful results on Der(Cq) in the chapter two of this paper. They are very essential to the proof of the final theorem. Finally, we study its derivation Lie algebra in the situation of d= 2 and d= 3 in the chapter three spectively, and show that when d= 2 it is isomorphic to g⊕(?) where (?)={D(u,0)}, but when d= 3 it is isomorphic to (?).