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We describe the structure and different features of Lie algebras in the Verlinde category, obtained as semisimplification of contragredient Lie algebras in characteristic p with respect to the adjoint action of a Chevalley generator. In particular, we construct a root system for these algebras that arises as a parabolic restriction of the known root system for the classical Lie algebra. This gives a lattice grading with simple homogeneous components and a triangular decomposition for the semisimplified Lie algebra. We also obtain a non-degenerate invariant form that behaves well with the lattice grading. As an application, we exhibit concrete new examples of Lie algebras in the Verlinde category. 

Abstract We show that over a perfect field, every non‐semisimple finite tensor category with finitely generated cohomology embeds into a larger such category where the tensor product property does not hold for support varieties. 

We show that over a perfect field, every non-semisimple finite tensor category with finitely generated cohomology embeds into a larger such category where the tensor product property does not hold for support varieties. 

We consider the finite generation property for cohomology of a finite tensor category C \mathscr {C} , which requires that the self-extension algebra of the unit \operatorname {Ext}^\text {\tiny ∙ }_\mathscr {C}(\mathbf {1},\mathbf {1}) is a finitely generated algebra and that, for each object V V in C \mathscr {C} , the graded extension group \operatorname {Ext}^\text {\tiny ∙ }_\mathscr {C}(\mathbf {1},V) is a finitely generated module over the aforementioned algebra. We prove that this cohomological finiteness property is preserved under duality (with respect to exact module categories) and taking the Drinfeld center, under suitable restrictions on C \mathscr {C} . For example, the stated result holds when C \mathscr {C} is a braided tensor category of odd Frobenius-Perron dimension. By applying our general results, we obtain a number of new examples of finite tensor categories with finitely generated cohomology. In characteristic 0 0 , we show that dynamical quantum groups at roots of unity have finitely generated cohomology. We also provide a new class of examples in finite characteristic which are constructed via infinitesimal group schemes. 

Abstract For a braided fusion category $$\mathcal{V}$$, a $$\mathcal{V}$$-fusion category is a fusion category $$\mathcal{C}$$ equipped with a braided monoidal functor $$\mathcal{F}:\mathcal{V} \to Z(\mathcal{C})$$. Given a fixed $$\mathcal{V}$$-fusion category $$(\mathcal{C}, \mathcal{F})$$ and a fixed $$G$$-graded extension $$\mathcal{C}\subseteq \mathcal{D}$$ as an ordinary fusion category, we characterize the enrichments $$\widetilde{\mathcal{F}}:\mathcal{V} \to Z(\mathcal{D})$$ of $$\mathcal{D}$$ that are compatible with the enrichment of $$\mathcal{C}$$. We show that G-crossed extensions of a braided fusion category $$\mathcal{C}$$ are G-extensions of the canonical enrichment of $$\mathcal{C}$$ over itself. As an application, we parameterize the set of $$G$$-crossed braidings on a fixed $$G$$-graded fusion category in terms of certain subcategories of its center, extending Nikshych’s classification of the braidings on a fusion category.