The main topic of my talk concerns constructions of "quantum" (or "q>1") analogues of the Delannoy category of Harman, Snowden, and Snyder. These categories can be defined by constructing relative measures on suitable oligomorphic groups in the sense of Harman and Snowden. In the "Borel" case, the oligomorphic group is the group of linear transformations of the free vector space over a finite field on the real numbers which preserve the canonical maximal flag. There is also a unipotent version which was suggested by P. Deligne. I will discuss various properties of these categories, including results about simple objects, and connection with the Delannoy category, resulting from a universal property of that category. I will also briefly mention my own formalism of "T-algebras," in which I first discovered the Borel q>1-Delannoy category.
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