Interpolations of monoidal categories and algebraic structures by invariant theory

Abstract

AbstractIn this paper we give a general construction of symmetric monoidal categories that generalizes Deligne’s interpolated categories, the categories introduced by Knop, and the recent TQFT construction of Khovanov, Ostrik, and Kononov. The categories we will consider are generated by an algebraic structure. In a previous work by the author a universal ring of invariants $$\mathfrak {U}$$ U for algebraic structures of a specific type was constructed. It was shown that any algebraic structure of this type in $${\text {Vec}}_K$$ Vec K gives rise to a character $$\chi :\mathfrak {U}\rightarrow K$$ χ : U → K . In this paper we consider algebraic structure in general symmetric monoidal categories, not only in $${\text {Vec}}_K$$ Vec K , and general characters on $$\mathfrak {U}$$ U . From any character $$\chi :\mathfrak {U}\rightarrow K$$ χ : U → K we construct a symmetric monoidal category $${\mathcal {C}}_{\chi }$$ C χ , 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 $$\chi $$ χ is good then $${\mathcal {C}}_{\chi }$$ C χ is abelian and semisimple, and that the set of good characters forms a K-algebra. We also show that the categories $${\mathcal {C}}_{\chi }$$ C χ contain all categories of the form $${\text {Rep}}(G)$$ Rep ( G ) , where G is reductive. The construction of $${\mathcal {C}}_{\chi }$$ C χ gives a way to interpolate algebraic structures, and also symmetric monoidal categories, in a way that generalizes Deligne’s categories $${\text {Rep}}(S_t)$$ Rep ( S t ) , $${\text {Rep}}({\text {GL}}_t(K))$$ Rep ( GL t ( K ) ) , and $${\text {Rep}}(\text {O}_t)$$ Rep ( O t ) . 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 $${\text {Rep}}({\text {Aut}}_{{\mathcal {O}}}(M))$$ Rep ( Aut O ( M ) ) where $${\mathcal {O}}$$ 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 $$S_t$$ S t , which was introduced by Knop.