Strict Triangle Derivation Lie Algebra Structure

In recent years,a new area in the representation theory has emerged.the theory of bounded modules for infinite-dimensional Lie algebras with a dense Zd-grading. The classical case n=1 includes the Kac-Moddy algebra and Virasoro algebra. Let be the ring of laurent polynomials in d≥2 commuting variables on the field of complex number C, let D=Der(A) be the Lie algebra of the derivations of A on a d-dimensional torus. Moreover, one of the most natural Lie algebras with a dense Z2-grading is the Lie algebra of the derivations on 2-dimensional torus, which is also called the Lie algebra of vector fields on a 2-dimensional torus. Fix the column vector space V=Cd over the field of complex numbers C with a standard basis{e1,e2,…,ed}. Let (·,·) be the bilinear form on V such that be a lattice in V. For denotetn=Let For and We denote foru∈Cd Then D (u, r)∈DerA. Let Der where And DerA has the Lie structure for u, v∈V and r.s∈Γ,whereIn Chapter 2 we study some characters of the strict triangular derivation Lie algebra: It is easy to check that (?)is a Lie algebra. We call (?) is a strict triangular derivation Lie algebra.In Chapter 3, we construct a class of representations of the strict triangular derivation Lie algebra, which has the representation space of the form C [t1±1,…, td±1]. We study the structure of the image modules of corresponding linear Lie algebra modules,under the Larsson functor Fα.Then we classify the irreducible modules.