Representations of the oriented skein category
Jonathan Brundan

TL;DR
This paper develops the representation theory of the oriented skein category, a key structure in knot theory, by analyzing its Grothendieck ring and constructing a graded lift through 2-representations, covering both semisimple and non-semisimple cases.
Contribution
It introduces a highest weight theory approach to the representation theory of $OS(z,t)$ and constructs a graded lift via 2-representations, extending understanding beyond semisimple cases.
Findings
Determined the Grothendieck ring for all parameter choices.
Constructed a graded lift as a 2-representation of a Kac-Moody 2-category.
Discussed the degenerate oriented Brauer category $OB( ext{delta})$.
Abstract
The oriented skein category is a ribbon category which underpins the definition of the HOMFLY-PT invariant of an oriented link, in the same way that the Temperley-Lieb category underpins the Jones polynomial. In this article, we develop its representation theory using a highest weight theory approach. This allows us to determine the Grothendieck ring of its additive Karoubi envelope for all possible choices of parameters, including the (already well-known) semisimple case, and all non-semisimple situations. Then we construct a graded lift of by realizing it as a 2-representation of a Kac-Moody 2-category. We also discuss the degenerate analog of , which is the oriented Brauer category .
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Taxonomy
TopicsAlgebraic structures and combinatorial models · Nonlinear Waves and Solitons · Advanced Combinatorial Mathematics
