2021/05/10 by Ehud Meir, Meir, Ehud · 1 citation
Mathematics · #14L24 #16R30 #18M05 #18M15 #57K16 #Advanced Topics in Algebra #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Quantum Algebra (math.QA)
paper · doi:10.48550/arxiv.2105.04622
openalex publication_date 2021/05/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In a previous work by the author it was shown that every finite dimensional algebraic structure over an algebraically closed field of characteristic zero K gives rise to a character K[X]aug→ K, where K[X]aug is a commutative Hopf algebra that encodes scalar invariants of structures. This enabled us to think of some characters K[X]aug→ K as algebraic structures with closed orbit. In this paper we study structures in general symmetric monoidal categories, and not only in VecK. We show that every character χ: K[X]aug→ K arises from such a structure, by constructing a category Cχ that is analogous to the universal construction from TQFT. We then give necessary and sufficient conditions for a given character to arise from a structure in an abelian category with finite dimensional hom-spaces. We call such characters good characters. We show that if χ is good then Cχ is abelian and semisimple, and that the set of good characters forms a K-algebra. This gives us a way to interpolate algebraic structures, and also symmetric monoidal categories, in a way that generalizes Deligne's categories Rep(St), Rep(GLt(K)), Rep(Ot), and also some of the symmetric monoidal categories introduced by Knop. We also explain how one can recover the recent construction of 2 dimensional TQFT of Khovanov, Ostrik, and Kononov, by the methods presented here. We give new examples, of interpolations of the categories Rep(AutO(M)) where O is a discrete valuation ring with a finite residue field, and M is a finite module over it. We also generalize the construction of wreath products with St, which was introduced by Knop.