Frobenius Algebras Associated with the \(\alpha \)-Induction for Equivariantly Braided Tensor Categories

Let G be a group. We give a categorical definition of the G-equivariant \(\alpha \) $\alpha$ -induction associated with a given G-equivariant Frobenius algebra in a G-braided multitensor category, which generalizes the \(\alpha \) $\alpha$ -induction for G-twisted representations of conformal nets. For a given G-equivariant Frobenius algebra in a spherical G-braided fusion category, we construct a G-equivariant Frobenius algebra, which we call a G-equivariant \(\alpha \) $\alpha$ -induction Frobenius algebra, in a suitably defined category called neutral double. This construction generalizes Rehren’s construction of \(\alpha \) $\alpha$ -induction Q-systems. Finally, we define the notion of the G-equivariant full centre of a G-equivariant Frobenius algebra in a spherical G-braided fusion category and show that it indeed coincides with the corresponding G-equivariant \(\alpha \) $\alpha$ -induction Frobenius algebra, which generalizes a theorem of Bischoff, Kawahigashi and Longo.