TL;DR
This paper extends the theory of representation categories to complex rank, defining analogs of classical categories for non-integer complex parameters and establishing foundational results and frameworks for future research.
Contribution
It introduces complex rank analogs of key representation categories and develops foundational tools for their study, expanding the scope of representation theory.
Findings
Defined complex rank analogs of category O and other representation categories
Developed a framework for studying these categories
Outlined directions for future research
Abstract
This paper is a sequel to arXiv:1401.6321. We define and study representation categories based on Deligne categories Rep(GL_t), Rep(O_t), Rep(Sp_2t), where t is any (non-integer) complex number. Namely, we define complex rank analogs of the parabolic category O and the representation categories of real reductive Lie groups and supergroups, affine Lie algebras, and Yangians. We develop a framework and language for studying these categories, prove basic results about them, and outline a number of directions of further research.
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