Reconstruction of tensor categories from their structure invariants

Abstract

In this paper, we study tensor (or monoidal) categories of finite rank over an algebraically closed field F. Given a tensor category C, we have two structure invariants of C: the Green ring (or the representation ring) r(C) and the Auslander algebra A(C) of C. We show that a Krull-Schmit abelian tensor category C of finite rank is uniquely determined (up to tensor equivalences) by its two structure invariants and the associated associator system of C. In fact, we can reconstruct the tensor category C from its two invariants and the associator system. More general, given a quadruple (R, A, φ, a) satisfying certain conditions, where R is a Z+-ring of rank n, A is a finite dimensional F-algebra with a complete set of n primitive orthogonal idempotents, φ is an algebra map from A ⊗F A to an algebra M(R, A) constructed from A and R, and a = {ai,j,l |1 6 i, j, l 6 n} is a family of “invertible” matrices over A, we can construct a Krull-Schmidt and abelian tensor category C over F such that R is the Green ring of C and A is the Auslander algebra of C. In this case, C has finitely many indecomposable objects (up to isomorphisms) and finite dimensional Hom-spaces. Moreover, we will give a necessary and sufficient condition for such two tensor categories to be tensor equivalent.