We point out that results of Shimizu on internal characters imply a usefulnon-semisimple variant of the categorical Verlinde formula for factorisablefinite tensor categories. When combined with results on pseudo-trace functionsby Miyamoto and Arike-Nagatomo, one can make a precise conjecture for anon-semisimple modular Verlinde formula which relates modular properties ofpseudo-trace functions and the product in the Grothendieck ring of thecorresponding vertex operator algebra.
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