Interpolated family of non-group-like simple integral fusion rings of Lie type

Abstract

This paper is motivated by the quest of a non-group irreducible finite index depth 2 maximal subfactor. We compute the generic fusion rules of the Grothendieck ring of Rep(PSL(2,[Formula: see text])), [Formula: see text] prime-power, by applying a Verlinde-like formula on the generic character table. We then prove that this family of fusion rings [Formula: see text] interpolates to all integers [Formula: see text], providing (when [Formula: see text] is not prime-power) the first example of infinite family of non-group-like simple integral fusion rings. Furthermore, they pass all the known criteria of (unitary) categorification. This provides infinitely many serious candidates for solving the famous open problem of whether there exists an integral fusion category which is not weakly group-theoretical. We prove that a complex categorification (if any) of an interpolated fusion ring [Formula: see text] (with [Formula: see text] non-prime-power) cannot be braided, and so its Drinfeld center must be simple. In general, this paper proves that a non-pointed simple fusion category is non-braided if and only if its Drinfeld center is simple; and also that every simple integral fusion category is weakly group-theoretical if and only if every simple integral modular fusion category is pointed.