ABSTRACT
Abstract We study the Temperley-Lieb algebra, central to the Jones polynomial invariant of knots and ensuing developments, from a novel point of view. We relate the Temperley-Lieb category to the categorical formulation of quantum mechanics introduced by Abramsky and Coecke as the basis for the development of high-level methods for quantum information and computation. We develop some structural properties of the Temperley-Lieb category, giving a simple diagrammatic description of epi-monic factorization, and hence of splitting idempotents. We then relate the Temperley-Lieb category to some topics in proof theory and computation. We give a direct, “fully abstract” description of the Temperley-Lieb category, in which arrows are just relations on discrete finite sets, with planarity being characterized by simple order-theoretic properties. The composition is described in terms of the “Geometry of Interaction” construction, originally introduced to analyze cut elimination in Linear Logic. Thus we obtain a planar version of Geometry of Interaction. Moreover, we get an explicit description of the free pivotal category on one self-dual object, which is easily generalized to an arbitrary generating category. Moreover, we show that the construction naturally lifts a dagger structure on the underlying category, thus exhibiting a key feature of the Abramsky-Coecke axiomatization. The dagger or “adjoint”, and the “complex conjugate”, acquire natural diagrammatic readings in this context. Finally, we interpret a
non-commutative lambda calculus (a variant of the Lambek calculus, widely used in computational linguistics) in the Temperley-Lieb category, and thus show how diagrammatic simplification can be viewed as functional computation.
