Centre of an algebra
The author's motivation for studying centers of algebras comes from rational conformal field theories (RCFTs). Since it has been known for quite a while that a lot of information about the chiral half of a RCFT is contained in a certain monoidal category, modular categories formulated by \textit{V. G. Turaev} [Quantum invariants of knots and 3-manifolds. 2nd revised ed. de Gruyter Studies in Mathematics 18. Berlin: Walter de Gruyter (2010; Zbl 1213.57002), Quantum invariants of knots and 3-manifolds. de Gruyter Studies in Mathematics. 18. Berlin: Walter de Gruyter (1994; Zbl 0812.57003)] were studied extensively by mathematicians and theoretical physicists. It was realized recently that the categories of modules of certain algebras in the chiral modular category of an RCFT correspond to a consistent set of its boundary conditions, while certain commutative algebras in the monoidal center of the chiral modular category describe the full RCFT [\textit{L. Kong} and \textit{I. Runkel}, Adv. Math. 219, No.~5, 1548--1576 (2008; Zbl 1156.18003)]. The transition from algebras in a modular category to commutative algebras in its monoidal center was studied by \textit{J. Fröhlich, J. Fuchs, I. Runkel} and \textit{C. Schweigert} [Adv. Math. 199, No.~1, 192--329 (2006; Zbl 1087.18006)] and \textit{L. Kong} and \textit{I. Runkel} [loc. cit.], where the construction uses whopping specific properties of algebras and modular categories. This paper presents a construction associating to an algebra in a monoidal category a commutative algebra (full center) in the monoidal center of the monoidal category. It is proved, by extending the definition to module categories over a monoidal category, that the full center is Morita invariant. It is shown also that the construction gives the right answer when applied to algebras in a modular category. The author concludes the paper by full centers of separable algebras in categories of group-graded vector spaces and categories of representations of a group.
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