Temperley-Lieb Algebras: from Cut-Elimination to Knot Theory via Quantum Mechanics
Samson Abramsky
Abstract
The Temperley-Lieb algebra plays a central role in the Jones
polynomial invariant of knots and links, and the extensive ensuing
developments relating topology and physics. It arises as a particular
form of strongly compact closed category, a structure which in recent
work with Bob Coecke we have used as the basis for an axiomatic
formulation of Quantum Mechanics, well adapted for Quantum
Information and Computation. We shall give an introductory overview
of these ideas, and then show how so-called GoI-style constructions
can be used to give the first ``fully-abstract'' (i.e. no quotients)
presentation of the Temperley-Lieb algebra, as well as of related
structures such as the Brauer algebra, which play an important role
in Representation Theory.