Braided Near-group Categories

A near-group category is an additively semisimple category with a product such that all but one of the simple objects is invertible. We classify braided structures on near-group categories, and give explicit numerical formulas for their associativity and commutativity morphisms. 1. Results We consider near-group categories; that is, semisimple monoidal categories, whose set of simples consists of a finite group G of invertible elements together with one noninvertible element m. We restrict to the case where m is not a summand in mm, and we will say that G is the underlying group for such a category. Our categories also have a ground ring R, and we assume throughout that R is an integral domain. Tambara and Yamagami [TY] have classified the possible monoidal structures on such categories. Their result is: 1.1 Theorem (Tambara-Yamagami). Monoidal near-group categories correspond to pairs (χ, τ) where χ is a nondegenerate, symmetric R-valued bicharacter of G, and τ is a square root of 1/|G| Our main result is the following theorem, which gives necessary and sufficient conditions under which these categories admit a balanced (tortile) braided structure, and parametrizes the distinct braidings. 1.2 Theorem. Suppose G is a finite abelian group, C is a monoidal near-group category with underlying group G, and R contains roots of unity of order 8|G|. Then (1) C admits a braiding if and only if G is an elementary abelian 2-group; that is, every element has order 2. (2) The nonequivalent braidings on C are in one-to-one correspondence with (n+ 1)-tuples (δ1, . . . , δn, ǫ), where ǫ = ±1 and δi = ±1 for all i, and n is the rank of G. (3) Each braiding of C has exactly two choices of twist morphisms compatible with it. Explicit computation of the commutativities further establishes