- Date
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Wednesday 18 December 2024, 10.00 - 10.30
- Abstract
- The Temperley-Lieb (TL) category is an algebraic structure that was introduced in lattice statistical mechanics and has since found applications in several areas including integrable systems, low dimensional topology, and representation theory. Famously, it has a diagrammatic realisation whereby morphisms are represented by (linear combinations of) embedded curves in a square considered up to isotopy and a finitising local relation. In this talk, I will present the construction of a skeletal diagram category which extends the TL category by including diagrams of embedded curves on non-orientable bounded surfaces. Such diagrams utilise handle decompositions for surfaces and are considered up to a handle-slide equivalence. A full set of monoidal generators is given which include the TL generators, a family of orientable genus one diagrams, which are the components of a braiding, and a and family of non-orientable diagrams, which are the components of a certain natural transformation. I will present some results and conjectures regarding finitising quotients of this category, motivated by representation theory.
