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In this paper we consider the Birman Wenzl algebras over an arbitrary field and prove that they are cellular in the sense of Graham and Lehrer. Furthermore, we determine for which parameters the Birman Wenzl algebras are quasi-hereditary. So the general theory of cellular algebras and quasi-hereditary algebras applies to Birman Wenzl algebras. As a consequence, we can determine all irreducible representations of the Birman Wenzl algebras by linear algebra methods. We prove also that the new Hecke algebras induced from Birman Wenzl algebras are Frobenius over a field (but not always cellular). Academic Press

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Let T be an infinitely generated tilting module of projective dimension at most one over an arbitrary associative ring A, and let B be the endomorphism ring of T. We prove that if T is good, then there exists a ring C, a homological ring epimorphism B→C and a recollement among the (unbounded) derived module categories 𝒟C of C, 𝒟B of B and 𝒟A of A. In particular, the kernel of the total left‐derived functor T⊗B𝕃‐ is triangle equivalent to the derived module category 𝒟C. Conversely, if T⊗B𝕃‐ admits a fully faithful left adjoint functor, then T is good. Moreover, if T arises from an injective ring epimorphism, then C is isomorphic to the coproduct of two relevant rings. In the case of commutative rings, the ring C can be strengthened as the tensor product of two commutative rings. Consequently, we produce a large variety of examples (from Dedekind domains and p‐adic number theory, or Kronecker algebra) to show that two different stratifications of the derived module category of a ring by derived module categories of rings may have completely different derived composition factors (even up to ordering and up to derived equivalence), or different lengths. This shows that the Jordan–Hölder theorem fails even for stratifications by derived module categories.

We give an axiomatic framework for studying the representation theory of towers of algebras. We introduce a new class of algebras, contour algebras, generalising (and interpolating between) blob algebras and cyclotomic Temperley-Lieb algebras. We demonstrate the utility of our formalism by applying it to this class.