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Current algebras, highest weight categories and quivers

Authors
Journal
Advances in Mathematics
0001-8708
Publisher
Elsevier
Publication Date
Volume
216
Issue
2
Identifiers
DOI: 10.1016/j.aim.2007.06.006
Keywords
  • Current Algebra
  • Highest Weight Category
  • Kirillov–Reshetikhin Modules
  • Path Algebra
Disciplines
  • Mathematics

Abstract

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.

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