Research Projects

Recent research has focused on understanding Weyl modules (local and global) for map Lie algebras, where g⊗A is a simple finite-dimensional Lie algebra and A is a commutative associative algebra with unit over complex numbers C. Global Weyl modules were initially defined for loop algebra as a maximal integrable highest module generated by a non-zero weight vector. The study was extended to g⊗A, showing that global Weyl modules are projective objects in suitable categories and their weight spaces are right modules for a certain commutative algebra. The definition of local Weyl modules was further generalized to the setting of equivariant map algebras (g⊗A)^{Γ}. Equivariant map algebras are Lie algebras of algebraic maps from a scheme to a target finite-dimensional Lie algebra that are equivariant with respect to the action of a finite group \Gamma, including twisted loop algebra and twisted current algebra. The key ingredient to this study was the notation of certain twisting and non-twisting functors that relate the representation theory of map algebra and equivariant map algebra. Local and global Weyl modules have been defined and studied for \g \otimes A and twisted loop superalgebras, but they are less explored than non-super settings. Weyl modules for Lie superalgebras have many analogues but have striking differences, such as the Borel Lie superalgebra of basic Lie superalgebra being no conjugate under the action of Weyl group. Kac-modules play an important role in the representation theory of these modules.