An invitation to V.F.R. Jones' planar algebras (Intelligence of Low-dimensional Topology)
Nobuya Sato · Institutional Repositories DataBase (IRDB) · 2010
This article is aimed to explain the basic ideas of planar al- gebras for topological applications in mind.We will start with some familiar diagrammatic algebras and will give some known examples of subfactor planar algebras without any knowledge of subfactor theory.We will briefly see some applications of (sub- factor) planar algebras to low dimensional topology. Introduction-diagrammatic algebras-In the last 25 years, we have witnessed that pictorial expression of some algebras is useful to construct link invariants and 3-manifold invariants.For instance, pictorial expressions of the elements in Temperley-Lieb algebras induce the Jones polynomials for links and those of so-called BMW algebras introduced by J. Murakami in [Mu] and Birman-Wenzl in [BW] independently induce the Kauffman polynomials.As the introduction of this article, let us recall the definitions of the Temperley-Lieb algebras and the BMW algebras and we will describe the diagrams for the generating elements.In this article, our ground field is always assumed to be the field of complex numbers $\mathbb{C}$ .Example 1 Temperley-Lieb algebras.Definition 1 Let $n$ be a natural number and $\delta$ be a non-zero complex number.The n-th $Tem$ perley-Lieb algebra $TL_{n}(\delta)$ is an algebra over $\mathbb{C}$ generated by the unit 1 and $E_{1},$ $\ldots,$ $E_{n-1}$ obeying the following relations: