Steffen König

Cellular Algebras: Inflations and Morita Equivalences

Cellular algebras have recently been introduced by Graham and Lehrer [ 5 , 6 ] as a convenient axiomatization of all of the following algebras, each of them containing information on certain classical algebraic or finite groups: group algebras of symmetric groups in any characteristic, Hecke algebras of type A or B (or more generally, Ariki Koike algebras), Brauer algebras, Temperley–Lieb algebras, ( q -)Schur algebras, and so on. The problem of determining a parameter set for, or even constructing bases of simple modules, is in this way reduced (but of course not solved in general) to questions of linear algebra. The present paper has two aims. First, we make explicit an inductive construction of cellular algebras which has as input data of linear algebra, and which in fact produces all cellular algebras (but no other ones). This is what we call ‘inflation’. This construction also exhibits close relations between several of the above algebras, as can be seen from the computations in [ 6 ]. Among the consequences of the construction is a natural way of generalizing Hochschild cohomology. Another consequence is the construction of certain idempotents which is used in the second part of the paper. The second aim is to study Morita equivalences of cellular algebras. Since the input of many of the constructions of representation theory of finite-dimensional algebras is a basic algebra, it is useful to know whether any finite-dimensional cellular algebra is Morita equivalent to a basic one by a Morita equivalence that preserves the cellular structure. It turns out that the answer is ‘yes’ if the underlying field has characteristic other than 2. However, there are counterexamples in the case of characteristic 2, or more generally for any ring in which 2 is not invertible. This also tells us that the notion of ‘cellular’ cannot be defined only in terms of the module category. However, in any characteristic we find some useful Morita equivalences which are compatible with cellular structures.

Cellular algebras and quasi-hereditary algebras: a comparison

Cellular algebras have been defined in a computational way by the existence of a special kind of basis. We compare them with quasi-hereditary algebras, which are known to carry much homological and categorical structure. Among the properties to be discussed here are characterizations of quasi-hereditary algebras inside the class of cellular algebras in terms of vanishing of cohomology and in terms of positivity of the Cartan determinant.

Cartan decompositions and BGG-resolutions

Necessary and sufficient criteria are given for the existence of BGG-resolutions (finite resolutions of modules by finite direct sums of Weyl modules) for simple modules over quasi-hereditary algebras, which have strong exact Borel subalgebras and strong Δ-subalgebras. Our main technical tool is the existence of Cartan decompositions for these algebras. The results apply to simple objects in the BGG-categoryO of a finite-dimensional semisimple complex Lie algebra and to finite dimensional simple rational modules over simply connected semisimple algebraic groups.