2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/221043We consider algebras in a modular tensor category C. If the trace pairing of an algebra A in C is non-degenerate we associate to A a commutative algebra Z(A), called the full centre, in a doubled version of the category C. We prove that two simple algebras with non-degenerate trace pairing are Morita-equivalent if and only if their full centres are isomorphic as algebras. This result has an interesting interpretation in two-dimensional rational conformal field theory; it implies that there cannot be several incompatible sets of boundary conditions for a given bulk theory.28 pages, several figures, version to appear in Adv. MathCategory TheoryHigh Energy Physics - TheoryQuantum Algebra16D90; 18D10; 81T40text