Abstract. The purpose of this paper is to show that the recent proof by Ganyushkin, Mazorchuk and Steinberg of the parametrisation of simple modules over finite semigroup algebras due to Clifford, Munn and Ponizovskiȋ carries over to twisted finite category algebras. We observe that the parametrisations of simple modules over Brauer algebras, Temperley-Lieb algebras, and Jones algebras due to Graham and Lehrer can be obtained as special cases of our main result. We further note that the notion of weights in the context of Alperin's weight conjecture extends to twisted finite category algebras.
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