A general framework for cluster tilting is set up by showing that any quotient of a triangulated category modulo a tilting subcategory (i.e., a maximal 1-orthogonal subcategory) carries an induced abelian structure. These abelian quotients turn out to be module categories of Gorenstein algebras of dimension at most one.
Highest weight categories arising in Lie theory are known to be associated with finite dimensional quasi-hereditary algebras such as Schur algebras or blocks of category O. An analogue of the PBW theorem will be shown to hold for quasi-hereditary algebras: Up to Morita equivalence each such algebra has an exact Borel subalgebra. The category F (∆) of modules with standard (Verma, Weyl, . . . ) filtration, which is exact, but rarely abelian, will be shown to be equivalent to the category of representations of a directed box. This box is constructed as a quotient of a dg algebra associated with the A ∞ -structure on Ext * (∆, ∆). Its underlying algebra is an exact Borel subalgebra.
Abstract. The dominant dimension of an algebra A provides information about the connection between A-mod and B-mod for B = eAe, a certain centralizer subalgebra of A. Well-known examples of such a situation are the connection (given by Schur-Weyl duality) between Schur algebras and group algebras of symmetric groups, and the connection (given by Soergel's 'Struktursatz') between blocks of the category O of a complex semisimple Lie algebra and the coinvariant algebra. We study cohomological aspects of such connections, in the framework of highest weight categories. In this setup we characterize the dominant dimension of A by the vanishing of certain extension groups over A, we determine the range of degrees, for which certain cohomology groups over A and over eAe get identified, we show that Ringel duality does not change dominant dimensions and we determine the dominant dimension of Schur algebras.
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