Reconstruction of tensor categories from their structure invariants

In this paper, we study tensor (or monoidal) categories of finite rank over an algebraically closed field $mathbb F$. Given a tensor category $mathcal{C}$, we have two structure invariants of $mathcal{C}$: the Green ring (or the representation ring) $r(mathcal{C})$ and the Auslander algebra $A(mathcal{C})$ of $mathcal{C}$. We show that a Krull-Schmit abelian tensor category $mathcal{C}$ of finite rank is uniquely determined (up to tensor equivalences) by its two structure invariants and the associated associator system of $mathcal{C}$. In fact, we can reconstruct the tensor category $mathcal{C}$ from its two invarinats and the associator system. More general, given a quadruple $(R, A, phi, a)$ satisfying certain conditions, where $R$ is a $mathbb{Z}_+$-ring of rank $n$, $A$ is a finite dimensional $mathbb F$-algebra with a complete set of $n$ primitive orthogonal idempotents, $phi$ is an algebra map from $Aotimes_{mathbb F}A$ to an algebra $M(R, A, n)$ constructed from $A$ and $R$, and $a={a_{i,j,l}|1< i,j,l<n}$ is a family of invertible matrices over $A$, we can construct a Krull-Schmidt and abelian tensor category $mathcal C$ over $mathbb{F}$ such that $R$ is the Green ring of $mathcal C$ and $A$ is the Auslander algebra of $mathcal C$. In this case, $mathcal C$ 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.