Title：Reconstruction of tensor categories from their invariants
Speaker：Prof. Huixiang Chen
From： Yangzhou University
Time：4:30pm-5:30pm, 10 Sept
Abstract：In this paper, we study the tensor categories of finite rank over an algebraically closed field
$\mathbb F$. Given a data $(R, A, \phi, a)$ with some conditions supposed on it, where $R$ is a $\mathbb{Z}_+$-ring of rank $n$ over $\mathbb Z$, $A$ is a finite dimensional $\mathbb F$-algebra with a full set of $n$ primitive orthogonal idempotents, $\phi$ is an algebra map from $A\otimes_{\mathbb F}A$ to an algebra $M(R, A, n)$ constructed by $A$ and $R$, and $a=\{a_{i,j,l}|1\< i,j,l\<n\}$ is a family of ``invertible matrices over $A$, we display a procedure to construct a Krull Schmidt and abelian tensor category $\mathcal C$ over $\mathbb F$ from $(R, A, \phi, a)$ 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 and finite dimensional Hom-spaces. We also give a sufficient and necessary condition for such two tensor categories to be tensor equivalent. Conversely, from any given Krull Schmidt and abelian tensor category $\mathcal C$ over $\mathbb F$ with finitely many indecomposable objects and finite dimensional Hom-spaces, one can get a data $(R, A, \phi, a)$ which satisfies all the conditions supposed in the procedure above such that $R$ and $A$ are the Green ring and the Auslander algebra of $\mathcal C$, respectively.
Moreover, it is shown that the new Krull Schmidt and abelian tensor category constructed from the data $(R, A, \phi, a)$ is tensor equivalent to the original category $\mathcal C$.
$\mathbb F$. Given a data $(R, A, \phi, a)$ with some conditions supposed on it, where $R$ is a $\mathbb{Z}_+$-ring of rank $n$ over $\mathbb Z$, $A$ is a finite dimensional $\mathbb F$-algebra with a full set of $n$ primitive orthogonal idempotents, $\phi$ is an algebra map from $A\otimes_{\mathbb F}A$ to an algebra $M(R, A, n)$ constructed by $A$ and $R$, and $a=\{a_{i,j,l}|1\< i,j,l\<n\}$ is a family of ``invertible matrices over $A$, we display a procedure to construct a Krull Schmidt and abelian tensor category $\mathcal C$ over $\mathbb F$ from $(R, A, \phi, a)$ 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 and finite dimensional Hom-spaces. We also give a sufficient and necessary condition for such two tensor categories to be tensor equivalent. Conversely, from any given Krull Schmidt and abelian tensor category $\mathcal C$ over $\mathbb F$ with finitely many indecomposable objects and finite dimensional Hom-spaces, one can get a data $(R, A, \phi, a)$ which satisfies all the conditions supposed in the procedure above such that $R$ and $A$ are the Green ring and the Auslander algebra of $\mathcal C$, respectively.
Moreover, it is shown that the new Krull Schmidt and abelian tensor category constructed from the data $(R, A, \phi, a)$ is tensor equivalent to the original category $\mathcal C$.
