The skein category of the annulus

Abstract

We construct the skein category S of the annulus and show that it is equivalent to the affine Temperley-Lieb category of Graham and Lehrer. It leads to a skein theoretic description of the extended affine Temperley-Lieb algebras. We construct an endofunctor of S that corresponds, on the level of tangle diagrams, to the insertion of an arc connecting the inner and outer boundary of the annulus. We use it to define and construct towers of extended affine Temperley-Lieb algebra modules. It allows us to construct a tower of modules acting on spaces of link patterns on the punctured disc which play an important role in the study of loop models. In case of trivial Dehn twist we show that the direct sum of the representation spaces of the link pattern tower defines a graded algebra that may be regarded as a relative version of the Roger-Yang skein algebra of arcs and links on the punctured disc. We also describe the link pattern tower in terms of fused extended affine Temperley-Lieb algebra modules.

Currents in the dilute $O(n=1)$ model

“…This allows us to relate systems of different sizes L and L − 1 to each other by the recursion relations. Similar equations hold for the dense loop model [35].…”

Towers of Solutions of qKZ Equations and Their Applications to Loop Models

“…The collection (H n ) n≥0 of extended affine Hecke algebras forms a tower of algebras with respect to algebra morphisms H n → H n+1 that arise as descendants of arc insertion morphisms B n → B n+1 for the groups B n of affine n-braids, cf. [1,3,15]. In this paper, we study families (f (n) ) n≥0 of solutions f (n) of qKZ equations taking values in H n -modules V n that are naturally compatible to the tower structure.…”

“…µ n : V n → V n+1 up to a quasi-constant factor, where V n+1 is regarded as an H n -module through the tower structure of (H n ) n≥0 . In the terminology of [3], the collection {(V n , µ n )} n≥0 of H nmodules V n and H n -intertwiners µ n : V n → V n+1 is a tower of extended affine Hecke algebra modules. From this perspective, towers of solutions of qKZ equations are naturally associated with towers of extended affine Hecke algebra modules.…”

“…In [3], the first and third authors constructed a family of module towers, called link pattern towers, which depends on a twist parameter v. The link pattern tower actually descends to a tower of extended affine Temperley-Lieb algebra modules. The representations V n are realized on spaces of link patterns on the punctured disc, which alternatively can be interpreted as the quantum state spaces for the dense O(τ ) loop models on the half-infinite cylinder (with n the circumference of the cylinder).…”

Cylinder topological quantum field theory: A categorical presentation of classical field theory and its symmetries

“…We use geometric ideas behind constructions in skein and braid theory [14][15] [16], the theory of Temperly-Lieb-type objects [19] [20], the classification theory of subfactors [21] [22] and in the foundational theory of Walker and Morrison's blob complex [23] to offer categorical interpretations of topological models describing local classical field theory and their symmetries. Precisely, we use a special case of Walker's cylinder category construction [24] to associate to any classical field theory satisfying the axioms of the local framework introduced in [1] and [2], a symmetric monoidal double functor from the symmetric monoidal double category of oriented cobordisms of the appropriate dimension with codimension 1 corners to a certain symmetric monoidal double category.…”