The Casimir Number And The Determinant Of A Fusion Category

Let C be a fusion category over an algebraically closed field k of arbitrary characteristic. Two numerical invariants of C, that is, the Casimir number and the determinant of C are considered in this paper. These two numbers are both positive integers and admit the property that the Grothendieck algebra Gr(C) circle times(Z) K over any field K is semisimple if and only if any of these numbers is not zero in K. This shows that these two numbers have the same prime factors. If moreover C is pivotal, it gives a numerical criterion that C is nondegenerate if and only if any of these numbers is not zero in k. For the case that C is a spherical fusion category over the field C of complex numbers, these two numbers and the Frobenius-Schur exponent of C share the same prime factors. This may be thought of as another version of the Cauchy theorem for spherical fusion categories.