For a multifusion category $ \mathcal{C} $ over an algebraically closed field it is known that $ \text{End}(\mathcal{1}) $ is a commutative semi-simple algebra. See, for example, Theorem 4.3.1 in [1]. In particular this justifies the name 'multifusion category' as $ \mathcal{C} $ then decomposes into $ \bigoplus_{ij} \mathcal{C}_{ij} $ where $ i $ and $ j $ range over the simple summands of $ \mathcal{1} $ and $ \mathcal{C}_{ii} $ is a fusion category.

My question is as follows: under what conditions does this also follow over an arbitrary field? In particular, is it enough to assume that every simple object $ S $ in $ \mathcal{C} $ satisfies $ \text{End}(\mathcal{S}) = \mathbb{K} $ i.e. that the category is split?

[1] Etingof, Pavel; Gelaki, Shlomo; Nikshych, Dmitri; Ostrik, Victor, Tensor categories, Mathematical Surveys and Monographs 205. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-2024-6/hbk). xvi, 343 p. (2015). ZBL1365.18001.


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