A Class Of Infinite-dimensional Lie Algebra

Let A := C [z₁^±1,…, z_d^±1] be the ring of laurent polynomial in d≥2 commuting variables on the field of complex number C, and let Der (A) be the Lie algebra of the derivations of A. For n = (n₁, n₂,…, n_d)^T∈Z^d, denote zⁿ = z₁^n₁z₂^n₂…z_d^n_d. Let D_i (n) - zⁿz_i ((?)/(?)_zi) , i = 1,…, d. We denote D (u, r) = (?)u_iD_i (r) for u∈C^d.In the Chapters 2 and 3 of this paper, we study the structure and representation of the triangular derivation Lie algebra:(?) := {D(u,r):u∈C^d,r∈Z^d such that u_ir_j = 0 when i < j} .It is given the characterization of the triangular derivation Lie algebra in chapter 2, and one shows that the triangular derivation Lie algebra it is perfect. In Chapter 3, we construct a class representation of the triangular derivation Lie algebra, which has the representation space of the form C [z₁^±1,…, z_d^±1] . We give the sufficient and necessary condition such that the representation is irreducible, we also study the structure of all nonzero proper submodule when the representation is not irreducible.In Chapter 4, we study the derivation Lie algebra of the following higher rank Virasoro-like algebra:g := span_C {D (u, r):r∈Z^d\ {0}, u∈Ker (r)} ,one shows that g isomorphis to the following skew derivation Lie algebra:(?) := span_C {D (u, r) : u∈Ker (r)} .We describe the automorphism groups Aut(g) and Aut(￡), respectively, of higher rank Virasoro-like algebra and skew derivation Lie algebra, it is obtained the following results:This generalizes the result of the correlative literature.