7 research outputs found
A categorical approach to Weyl modules
Global and local Weyl Modules were introduced via generators and relations in
the context of affine Lie algebras in a work by the first author and Pressley
and were motivated by representations of quantum affine algebras. A more
general case was considered by Feigin and Loktev by replacing the polynomial
ring with the coordinate ring of an algebraic variety. We show that there is a
natural definition of the local and global modules via homological properties.
This characterization allows us to define the Weyl functor from the category of
left modules of a commutative algebra to the category of modules for a simple
Lie algebra. As an application we are able to understand the relationships of
these functors to tensor products, generalizing previous results. Finally an
analysis of the fundamental Weyl modules proves that the functors are not left
exact in general, even for coordinate rings of affine varieties.Comment: 29 page
Demazure Filtrations of Tensor Product Modules and Character Formula
We study the structure of the finite-dimensional representations of
, the current Lie algebra type of , which are obtained
by taking tensor products of special Demazure modules. We show that these
representations admit a Demazure flag and obtain a closed formula for the
graded multiplicities of the level 2 Demazure modules in the filtration of the
tensor product of two local Weyl modules for . Furthermore,
we derive an explicit expression for graded character of the tensor product of
a local Weyl module with an irreducible module. In
conjunction with the results of \cite{MR3210603}, our findings provide evidence
for the conjecture in \cite{9} that the tensor product of Demazure modules of
levels m and n respectively has a filtration by Demazure modules of level m +
n
Local Weyl modules for equivariant map algebras with free abelian group actions
Suppose a finite group acts on a scheme X and a finite-dimensional Lie
algebra g. The associated equivariant map algebra is the Lie algebra of
equivariant regular maps from X to g. Examples include generalized current
algebras and (twisted) multiloop algebras. Local Weyl modules play an important
role in the theory of finite-dimensional representations of loop algebras and
quantum affine algebras. In the current paper, we extend the definition of
local Weyl modules (previously defined only for generalized current algebras
and twisted loop algebras) to the setting of equivariant map algebras where g
is semisimple, X is affine of finite type, and the group is abelian and acts
freely on X. We do so by defining twisting and untwisting functors, which are
isomorphisms between certain categories of representations of equivariant map
algebras and their untwisted analogues. We also show that other properties of
local Weyl modules (e.g. their characterization by homological properties and a
tensor product property) extend to the more general setting considered in the
current paper.Comment: 18 pages. v2: Minor correction