9,747 research outputs found

### 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 $\mathfrak g$-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

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