Current algebras, highest weight categories and quivers

By Vyjayanthi Chari, Jacob Greenstein. 2006

60 citations recorded by the source corpus.

Abstract 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 parametrized by the affine weight lattice. We show that with respect to a suitable refinement of the standard ordering on the affine 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 an interval closed finite subset of the weight lattice. Finally, we prove that there is a large number of interesting quivers of finite, affine and star-shaped type, as well as tame quasi-hereditary algebras, that arise from our study.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/51f781480aa59d97793662b95ca0861394fb9fd8

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

A categorical approach to Weyl modules (2009, methodology): “This approach (as also the approach in [ CG1 ], [CG2]) is motivated by the methods used to study another well–known category in representation theory: the BGG-category O for semi–simple Lie algebras.”
BGG reciprocity for current algebras (2011, background): “per should also be true for the loop algebra. We describe our results in some detail. If P+ denotes the set of dominant integral weights of gand Z denotes the ring of integers, then it is easily seen [6] that the isomorphism classes of simple objects of Igr are indexed by P+ ×Z. For (λ,r) ∈ P+ ×Z, t”