Non-vanishing of the dimension of the invariant subalgebra A^D

Determine whether the pivotal dimension d(A^D) is nonzero for the invariant subobject A^D obtained as the image of the idempotent ρ(e_D) in End_C(A), where C is a modular tensor category, A is a condensable algebra in C, C_A is the fusion category of right A-modules, D is a fusion subcategory of C_A containing the local A-modules C, and ρ: K(C_A) → End_C(A) is the fusion action defined via generalized Frobenius–Schur indicators. In particular, ascertain whether d(A^D) ≠ 0 holds whenever D has positive (Frobenius–Perron) dimension, so that A^D is automatically condensable without additional assumptions.

Background

Given a condensable algebra A in a modular tensor category C, the authors define a fusion action ρ of the Grothendieck algebra K(C_A) on A. For a fusion subcategory D ⊆ C_A containing the local A-modules C, they form the central idempotent e_D and define A^D to be the image of ρ(e_D) in End_C(A).

To identify A^D as a condensable subalgebra, the proof requires that its pivotal dimension d(A^D) be nonzero. While this is immediate in the pseudounitary setting used later for the bijection, the nonvanishing of d(A^D) is not established in general, and the authors explicitly flag this as unclear. They note that it “has to be” nonzero if D has positive dimension, but leave this unproven at that point.