Finite, connected, semisimple, rigid tensor categories are linear

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 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.