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# Mathematical Research Letters

## Volume 10 (2003)

### Number 4

### Finite, connected, semisimple, rigid tensor categories are linear

Pages: 411 – 421

DOI: http://dx.doi.org/10.4310/MRL.2003.v10.n4.a1

#### Author

#### Abstract

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.