In this paper, we will study the notion of a Frobenius ⋆-algebra and prove some orthogonality relations for the irreducible characters of a Frobenius ⋆-algebra. Then we will study [Formula: see text]-graded Frobenius ⋆-algebras and prove some twisted orthogonality relations for them.

For a finite group G, Turaev introduced the notion of a braided G-crossed fusion category. The classification of braided G-crossed extensions of braided fusion categories was studied by Etingof, Nikshych and Ostrik in terms of certain group cohomological data. In this paper we will define the notion of a G-crossed Frobenius ⋆-algebra and give a classification of (strict) G-crossed extensions of a commutative Frobenius ⋆-algebra R equipped with a given action of G, in terms of the second group cohomology H 2 (G, R × ). Now suppose that B is a non-degenerate braided fusion category equipped with a braided action of a finite group G. We will see that the associated G-graded fusion ring is in fact a (strict) G-crossed Frobenius ⋆-algebra. We will describe this G-crossed fusion ring in terms of the classification of braided G-actions by Etingof, Nikshych, Ostrik and derive a Verlinde formula to compute its fusion coefficients.

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