In order to investigate the representation theory of a vertex algebra , a fruitful strategy is to look at the properties of its -algebra . This Poisson algebra reflects interesting properties of the vertex algebra and is often easier to handle than the vertex algebra itself. In this talk, we are interested in studying vertex algebras in a closed monoidal category, and in providing a description of the dual versions of and when those exist. We introduce vertex algebras graded by an abelian group and explain how to "dualize" the definition to obtain a graded vertex coalgebra. This leads to the notion of the -coalgebra of a vertex coalgebra. We will describe its properties and show that the duality vertex algebra / vertex coalgebra passes down to a duality -algebra / -coalgebra. We will also explain how this dualities carry on to the respective modules / comodules.
The Effros-Shen algebra corresponding to an irrational number is classically described by an inductive sequence of direct sums of matrix algebras determined by the continued fraction expansion of . In recent work, Mitscher and Spielberg present an Effros-Shen algebra as the -algebra of a category of paths -- a generalization of a directed graph — also corresponding to the continued fraction expansion of . With this approach, the algebra is realized as the inductive limit of a sequence of infinite-dimensional (rather than finite-dimensional) subalgebras. In this talk, we will see approaches to constructing spectral triples on an Effros-Shen algebra, both in the classical picture (based on work of Adams, Aguilar, Ayala, Knight, and Marple), and in terms of the category of paths presentation, in both cases drawing on a construction of Christensen and Ivan. This is joint work with Konrad Aguilar and Jack Spielberg.
