Centre of an algebra
Alexei Davydov
TL;DR
This work develops a general theory of centers for algebras in monoidal categories by introducing the full centre $Z(A)$ as a commutative algebra in the monoidal centre ${\cal Z}({\cal C})$ via a universal property. It proves Morita invariance by extending the construction to module categories and connecting it to left centres through adjunctions, with explicit formulations in braided and modular settings. In the modular case the full centre is computed as $Z(A)=C_l(\iota_+(A)\otimes\tilde{R})$, tying algebraic centers to coends and induction data intrinsic to modular categories. The paper provides concrete group theoretic examples in ${\cal C}(G)$ and ${\cal R}ep(G)$, illustrating how full centres recover familiar objects like skew group algebras and centres of group algebras, and it relates RCFT structures to a universal categorical framework. Overall, it generalizes and unifies center constructions across monoidal, braided, and modular contexts with Morita invariance as a central guiding principle.
Abstract
Motivated by algebraic structures appearing in Rational Conformal Field Theory we study a construction associating to an algebra in a monoidal category a commutative algebra ({\em full centre}) in the monoidal centre of the monoidal category. We establish Morita invariance of this construction by extending it to module categories. As an example we treat the case of group-theoretical categories.