Abstract
Graham and Lehrer (1998) introduced a Temperley-Lieb category \ctl whose objects are the non-negative integers and the morphisms in \Hom(n,m) are the link diagrams from nn to mm nodes. The Temperley-Lieb algebra \tl n is identified with \Hom(n,n). The category \ctl is shown to be monoidal. We show that it is also a braided category by constructing explicitly a commutor. A twist is also defined on \ctl. We introduce a module category \modtl whose objects are functors from \ctl to \mathsf{Vect}_{\mathbb C}𝖵𝖾𝖼𝗍ℂ and define on it a fusion bifunctor extending the one introduced by Read and Saleur (2007). We use the natural morphisms constructed for \ctl to induce the structure of a ribbon category on \modtl(\beta=-q-q^{-1}), when qq is not a root of unity. We discuss how the braiding on \ctl and integrability of statistical models are related. The extension of these structures to the family of dilute Temperley-Lieb algebras is also discussed.
Highlights
The family of the Temperley-Lieb algebras was cast into a categorical framework by Graham and Lehrer [1] in 1998
For several XXZ and loop models [6,7], the Hamiltonian is first defined as an element of a Temperley-Lieb algebra TLn, or one of its generalizations, and the actual linear operator is obtained as the representative of this element in some representations over the algebra
For the TemperleyLieb algebra, such a fusion product −1 ×f −2 was introduced by Read and Saleur [2] and computed in many cases by Gainutdinov and Vasseur [3] and Belletête [4]
Summary
The (original) family of the Temperley-Lieb algebras was cast into a categorical framework by Graham and Lehrer [1] in 1998. Still later this product was computed between several families of modules by Gainutdinov and Vasseur [3], and Belletête [4] Their recent results (obtained in 2012 and 2015 respectively) lead to natural questions: how can one define the module category over Graham and Lehrer’s category? For the TemperleyLieb algebra, such a fusion product −1 ×f −2 was introduced by Read and Saleur [2] and computed in many cases by Gainutdinov and Vasseur [3] and Belletête [4] It is associative and commutative, and the braiding gives the isomorphism between M ×f N and N ×f M.
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