Filtrations and completions of certain positive level modules of affine algebras

We define a filtration indexed by the integers on the tensor product of an integrable highest weight module and a loop module for a quantum affine algebra. We prove that the filtration is either trivial or strictly decreasing and give sufficient conditions for this to happen. In the first case we prove that the module is irreducible and in the second case we prove that the intersection of all the modules is zero, thus allowing us to define the completed tensor product. In certain special cases, we identify the subsequent quotients of filtration. These are certain highest weight integrable modules and the multiplicity and the highest weight are the same as that obtained by decomposing the tensor product of the highest weight crystal bases with the crystal bases of a loop module

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

Minimal Affinizations of Representations of Quantum Groups: the simply--laced case

We continue our study of minimal affinizations for algebras of type D, E.Comment: 25 page

Current algebras, highest weight categories and quivers

We study the category of graded finite-dimensional representations of the polynomial current algebra associated to a simple Lie algebra. We prove that the category has enough injectives and compute the graded character of the injective envelopes of the simple objects as well as extensions between simple objects. The simple objects in the category are parametized by the affine weight lattice. We show that with respect to a suitable refinement of the standard ordering on affine the weight lattice the category is highest weight. We compute the Ext quiver of the algebra of endomorphisms of the injective cogenerator of the subcategory associated to a interval closed finite subset of the weight lattice. Finally, we prove that there is a large number of interesting quivers of finite, affine and tame type that arise from our study. We also prove that the path algebra of star shaped quivers are the Ext algebra of a suitable subcategory.Comment: AMSLaTeX, 25 page

Quivers with relations arising from Koszul algebras of g\mathfrak g-invariants

Let $\mathfrak g$ be a complex simple Lie algebra and let $\Psi$ be an extremal set of positive roots. One associates with $\Psi$ an infinite dimensional Koszul algebra \bold S_\Psi^{\lie g} which is a graded subalgebra of the locally finite part of ((\bold V)^{op}\tensor S(\lie g))^{\lie g}, where $\bold V$ is the direct sum of all simple finite dimensional \lie g-modules. We describe the structure of the algebra \bold S_\Psi^{\lie g} explicitly in terms of an infinite quiver with relations for \lie g of types $A$ and $C$. We also describe several infinite families of quivers and finite dimensional algebras arising from this construction.Comment: 49 pages, AMSLaTeX+amsref