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Simple algebras and exact module categories
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We verify a conjecture of Etingof and Ostrik, stating that an algebra object in a finite tensor category is exact if and only if it is a finite direct product of simple algebras. Towards that end, we introduce an analogue of the Jacobson radical of an algebra object, similar to the Jacobson radical of a finite-dimensional algebra. We give applications of our main results in the context of incompressible finite symmetric tensor categories.
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Cited by 2 Pith papers
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Clifford and Weyl algebras in symmetric tensor categories
For a symplectic object V in a Frobenius exact symmetric tensor category with finite symmetric algebra, the Weyl algebra A(V) is Azumaya, and the resulting symplectic Witt group is described by Stiefel-Whitney classes...
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Relative Invertibility and Full Dualizability of Finite Braided Tensor Categories
A finite braided tensor category is fully dualizable in the Morita 4-category of braided pre-tensor categories whenever its symmetric center is separable.
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