Finite, connected, semisimple, rigid tensor categories are linear

Fusion categories are fundamental objects in quantum algebra, but their definition is narrow in some respects. By definition a fusion category must be $k$-linear for some field $k$, and every simple object $V$ is strongly simple, meaning that $\mathrm{End}(V) = k$. We prove that linearity follows automatically from semisimplicity: Every connected, finite, semisimple, rigid, monoidal category $\C$ is $k$-linear and finite-dimensional for some field $k$. Barring inseparable extensions, such a category becomes a multifusion category after passing to an algebraic extension of $k$. The proof depends on a result in Galois theory of independent interest, namely a finiteness theorem for abstract composita.

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Published 1 January 2003