1-quasi-hereditary algebras

Motivated by the structure of the algebras associated to the blocks of the BGG-category O we define a subclass of quasi-hereditary algebras called 1-quasi-hereditary. Many properties of these algebras only depend on the defining partial order. In particular, we can determine the quiver and the form of the relations. Moreover, if the Ringel dual of a 1-quasi-hereditary algebra is also 1-quasi-hereditary, then the structure of the characteristic tilting module can be computed.Comment: 20 pages, examples and some statements are removed and will appear in a separate file. Some proofs are rewritte

Quasi-hereditary algebras and generalized Koszul duality

We present an easily applicable sufficient condition for standard Koszul algebras to be Koszul with respect to $\Delta$. If a quasi-hereditary algebra \L is Koszul with respect to $\Delta$, then \L and the Yoneda extension algebra of $\Delta$ are Koszul dual in a sense explained below, implying in particular that their bounded derived categories of finitely generated graded modules are equivalent. We also prove that the extension algebra of $\Delta$ is Koszul in the classical sense.Comment: This is a revised and updated version of the last section of arXiv:1007.328

The Global Dimension of Schur Algebras for GL2 and GL3

AbstractWe first define the notion of good filtration dimension and Weyl filtration dimension in a quasi-hereditary algebra. We calculate these dimensions explicitly for all irreducible modules in SL2 and SL3. We use these to show that the global dimension of a Schur algebra for GL2 and GL3 is twice the good filtration dimension. To do this for SL3, we give an explicit filtration of the modules ∇(λ) by modules of the form ∇(μ)F⊗L(ν) where μ is a dominant weight and ν is p-restricted