In this diagram, G is a connected complex semisimple group of adjoint type with Lie algebra g. We fix a Borel subgroup B ⊂ G, write b = Lie B ⊂ g for the corresponding Borel subalgebra, and n for the nilradical of b. Let Ñ := G×B n be the Springer resolution, and CohG×C ∗ (Ñ ) the abelian category ofG×C∗-equivariant coherent sheaves on Ñ , where the group G acts on Ñ by conjugation and C∗ acts ...

We study the abelian category W (p)-mod of modules over the triplet W algebra W (p). We construct the projective covers P± s of all the simple objects X± s , 1 ≦ s ≦ p, in the category W (p)-mod. By using the structure of these projective modules, we show that W (p)-mod is a category which is equivalent to the abelian category of the finite-dimensional modules for the restricted quantum group Ū...

We consider representations of quivers in arbitrary categories and twisted representations of quivers in arbitrary tensor categories. We show that if A is an abelian category, then the category of representations of a quiver in A is also abelian, and that the category of twisted linear representations of a quiver is equivalent to the category of linear (untwisted) representations of a different...

The general notion of what is now called Hall algebra or Ringel-Hall algebra is an algebra defined out of a certain type of category, a "finitary abelian category". Finitary abelian categories are, roughly speaking, categories where the notions of exact sequences make sense and such that for any pair of objects A,B in the category in question, we have # Hom(A,B) <∞ and # Ext1(A,B) <∞. The Hall ...

If A and B are Abelian groups, then Hom(A,B) is also an Abelian group under pointwise addition of functions. In this section we will see how Hom gives rise to classes of functors. Let A denote the category of Abelian groups. A covariant functor T : A → A associates to every Abelian group A an Abelian group T (A), and for every homomorphism f : A → B a homomorphism T (f) : T (A)→ T (B), such tha...

A tag module is a generalization, in any abelian category, of a torsion abelian group. The theory of such modules is developed, it is shown that countably generated tag modules are simply presented, and that Ulm's theorem holds for simply presented tag modules. Zippin's theorem is stated and proved for countably generated tag modules. 1. TAG-modules In the theory of torsion abelian groups, a di...