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Title: | Reconstruction of tensor categories from their structure invariants | Authors: | Chen, Huixiang ZHANG, Yinhuo |
Issue Date: | 2020 | Publisher: | BELGIAN MATHEMATICAL SOC TRIOMPHE | Source: | Bulletin of the Belgian Mathematical Society Simon Stevin (Printed), 27 (2) , p. 245 -279 | 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. | Keywords: | Green ring;Auslander algebra;associator;tensor category | Document URI: | http://hdl.handle.net/1942/31932 | Link to publication/dataset: | https://projecteuclid.org/euclid.bbms/1594346417 | ISSN: | 1370-1444 | e-ISSN: | 2034-1970 | ISI #: | WOS:000549353400005 | Rights: | 2020 Project Euclid. | Category: | A1 | Type: | Journal Contribution | Validations: | ecoom 2021 |
Appears in Collections: | Research publications |
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