Virtual algebraic Lie theory: Tilting modules and Ringel duals for blob algebras

In this paper we construct a representation of the blob algebra [22] over a ring allowing base change to every interesting (i.e. non–semisimple) specialisation which, in quasihereditary specialisations, passes to a full tilting module. The Temperley–Lieb algebras are a tower T0(q) ⊂ T1(q) ⊂ .. of one–parameter finite dimensional algebras [28], each with a basis independent of q. These algebras are quasihereditary [3, 8] except in case q+q = 0. Accordingly one may in principle construct tilting modules, full tilting modules, and corresponding Ringel duals. In fact, if V is a free module of rank 2 over the ground ring then Tn(q) has an action on V , and it is straightforward to show (see later) that V ⊗n is a full tilting module in the quasihereditary cases. Since V ⊗n exists over the ground ring, the Ringel dual can be constructed without having to pick a specialisation. The cases of n finite of this dual are a nested sequence of quotients of the quantum group Uqsl2 [17, 11]. This q–deformable duality and glorious limit structure [16] (more usually observed with Uqsl2 as the starting point) provides the mechanism for massive exchange of representation theoretic information between the two sides [26, 10, 13, 4, 18]. In particular the weight theory of Uqsl2 controls the representation theory of Tn(q) for all n simultaneously (as localisations of a global limit). The blob algebras are a tower b0 ⊂ b1 ⊂ .. of two–parameter finite dimensional algebras (and bn ⊃ Tn(q)). They are quasihereditary except at a finite set of parameter values. Accordingly one may in principle construct tilting modules and so on. Ab initio one would have to expect such a construction to depend on the specialisation, as indecomposable tilting modules do [24]. On the other hand, it turns out [23] that bn has an action on V , and in this paper we show that V ⊗2n is a full tilting module in the quasihereditary cases. Historically, Tn(q) and Uqsl2 were studied extensively separately, before the full tilting module/Ringel duality connection was known, but if one side, and the appropriate full tilting module, had been discovered first, the passage to the Ringel dual would rightly have been regarded as quite a significant spin–off! The bn tilting property of V ⊗2n is a striking result, in as much as it places us in a position analogous to this (as it were, before the discovery of quantum groups).