Classification of irreducible representations of Lie algebra of vector fields on a torus

We solve a long standing problem of the classification of all simple modules with finite-dimensional weight spaces over Lie algebra of vector fields on n-dimensional torus for any n. This generalizes the classical result of O.Mathieu on simple weight modules for the Virasoro algebra (n = 1). Every such module is either of a highest weight type or is a quotient of a module of tensor fields on a torus, which was conjectured by Eswara Rao

Classification of simple cuspidal modules for solenoidal Lie algebras

We give a new conceptual proof of the classification of cuspidal modules for the solenoidal Lie algebra. This classification was originally published by Y. Su in 2001. Our proof is based on the theory of modules for the solenoidal Lie algebras that admit a compatible action of the commutative algebra of functions on a torus

Lie algebras of vector fields on smooth affine varieties

We reprove the results of Jordan [18] and Siebert [30] and show that the Lie algebra of polynomial vector fields on an irreducible affine variety X is simple if and only if X is a smooth variety. Given proof is self-contained and does not depend on papers mentioned above. Besides, the structure of the module of polynomial functions on an irreducible smooth affine variety over the Lie algebra of vector fields is studied. Examples of Lie algebras of polynomial vector fields on an N-dimensional sphere, non-singular hyperelliptic curves and linear algebraic groups are considered

Representations of Lie algebras of vector fields on affine varieties

For an irreducible affine variety X over an algebraically closed field of characteristic zero we define two new classes of modules over the Lie algebra of vector fields on X—gauge modules and Rudakov modules, which admit a compatible action of the algebra of functions. Gauge modules are generalizations of modules of tensor densities whose construction was inspired by non-abelian gauge theory. Rudakov modules are generalizations of a family of induced modules over the Lie algebra of derivations of a polynomial ring studied by Rudakov [23]. We prove general simplicity theorems for these two types of modules and establish a pairing between them