On Structure Of Some Infinite Dimensional Lie Algebras Over The Quantum Torus

The extended affine Lie algebras (EALA for short) are an important class of Lie algebras which includes the well-known Kac-Moody Lie algebras of finite and affine type. These kinds of Lie algebras were first introduced in the paper [H-KT], and called the quasisimple Lie algebras. In [BGK], the authors studied the quasisimple Lie algebras of simply laced cases. In the book [AABGP], the authors used the concept of a semilattice to classify the extended affine root systems, and they providedconcrete realization of EALA's as well. Different from the affine Kac-Moody Lie algebras,the coordinate algebras of EALA's not only can be the Laurent polynomial algebras of multiple variables, but also can be alternative algebras, Jordan algebras and the quantum torus.The quantum torus, which are the non-commutative generalization of the Laurent polynomial rings, have been studied by many researchers, such as [P],[KPS],[McP]. From the quantum torus C_q and its derivation algebra Der(C_q), one can construct many important Lie algebras. For instance, the authors in [BGK] realized the EALA's of type A by using the quantum torus along with Lie subalgebras of Der(C_q). In this thesis, we mainly discuss the structure properties of some infinite dimensional Lie algebras, which are the extension of the skew derivation Lie algebras arising from the quantum torus in two variables.Let R² be the 2-dimensional Euclidean space. Z² is a lattice in R².Let f: Z²Ã—Z²â†’C^* be the map defined by f(a,b) = q^{a₂b₁-a₁b₂},where a = (a₁,a₂),b= (b₁,b₂)∈Z².The additive subgroup rad(f)={a∈Z²|f(a,b)= l,(?)b∈Z²} is called the radical of f.Let q∈C^*,and C_q:=C_q[(?),(?)] be the quantum torusof rank two associated to the matrix (?).It is an associative algebra withunity. The subalgebra of Der(C_q)B=span_C{x^a(a₂d₁-a₁d₂),(0,0)≠a∈rad(f)}(?)span_C{adx^r,r∈Z²,r(?)rad(f)}is called the skew derivation Lie algebra over C_q,where d_i,i=1,2 are the degree derivations. If q=1 or q is not a root of unity, then B is isomorphic to the Virasoro-like algebra or the q-analog of the Virasoro-like algebra respectively. Let (?)(q) = B(?)C_q,the semi-direct product of the Lie algebras B and C_q,where C_q is an ideal of (?)(q).This Lie algebra can be viewed as a generalization of the rank-one Heisenberg-Virasoro algebra. Denote L(q)=[(?)(q),(?)(q)], then we have (?)(q)=L(q)(?)C,where C is the center of (?)(q).In this thesis, I first determine the automorphism group, the derivation algebra, the covering of the Lie algebra L(q), where q≠1 is a p-th primitive root of unity. Next, I compute the Leibniz second cohomology group as well as the invariant symmetric bilinear form of B. The main contents are arranged as follows:In chapter one, we determine the automorphism group of L(q), we get thatAs a corollary, we obtain the automorphism group of (?)(q). These results generalize those obtained in [XLT] and [ZhT].In chapter two, we compute the derivations from L{q) to its adjoint module L(q).The result is Der(L(q))=Innder(L(q)) (?) Outder(L(q)), where Outder(L(q)) is of 5-dimensional.In chapter three, we give the universal covering (?) of L(q) by showing that the covering central extension (?) is determined by four linearly independent 2-cocycles of L(q) with values in C.In chapter four, we study the Leibniz second cohomology group H L²(B, C) of the skew derivaion Lie algebra B, and find that it is equal to the second cohomology group of B. Moreover, we show that the vector space consisting of the invariant symmetric bilinear forms of B is one dimensional. As a corollary, we obtain that H L²(B, C) = H²(B,C) by applying a result in [HPL].