Construction of Deligne Categories through ultrafilters and its applications

Abstract

The present thesis is concerned with the study of Deligne categories and their application to various representation-theoretic problems. The lens that is used to view Deligne categories in this study is the one of ultrafilters and ultraproducts. As will be shown in our work, this approach turns out to be a very powerful one. Especially if one wants to solve such representation-theoretic problems as presented by P.Etingof in his papers on "Representation theory in complex rank" ([13, 14]). The results are presented in two parts. In the first one (Chapters 2 and 3) an introduction to the theory of ultrafilters is given, and the construction of the Deligne categories through ultrafilters is presented. This also allows us to understand how one can make sense of Deligne categories as a limit in rank and characteristic. The later part of the text describes two applications of this construction to actual representation-theoretic problems. In Chapter 4 the full classification of simple commutative, associative and Lie algebras in Rep(𝑆𝜈) for 𝜈 /∈ Z≥0 is stated and proven. The second application, the construction of deformed double current algebras as a space of endomorphisms of a certain ind-object of Rep(𝑆𝜈), is contained in Chapter 5. There it is also proven that this construction agrees with Guay’s deformed double current algebra of type 𝐴 if the rank 𝑟 ≥ 4 (Guay’s algebra is presently only defined for such rank), and the presentation by generators and relations for the case of 𝑟 = 1 is given.