Suppose V^G is the fixed-point vertex operator subalgebra of a compact group G acting on a simple abelian intertwining algebra V. We show that if all irreducible V^G-modules contained in V live in some braided tensor category of V^G-modules, then they generate a tensor subcategory equivalent to the category Rep G of finite-dimensional representations of G, with associativity and braiding isomorphisms modified by the abelian 3-cocycle defining the abelian intertwining algebra structure on V. Additionally, we show that if the fusion rules for the irreducible V^G-modules contained in V agree with the dimensions of spaces of intertwiners among G-modules, then the irreducibles contained in V already generate a braided tensor category of V^G-modules. These results do not require rigidity on any tensor category of V^G-modules and thus apply to many examples where braided tensor category structure is known to exist but rigidity is not known; for example they apply when V^G is C_2-cofinite but not necessarily rational. When V^G is both C_2-cofinite and rational and V is a vertex operator algebra, we use the equivalence between Rep G and the corresponding subcategory of V^G-modules to show that V is also rational. As another application, we show that a certain category of modules for the Virasoro algebra at central charge 1 admits a braided tensor category structure equivalent to Rep SU(2), up to modification by an abelian 3-cocycle.

Nakaoka and Palu introduced the notion of extriangulated categories by extracting the similarities between exact categories and triangulated categories. In this paper, we study cotorsion pairs in a Frobenius extriangulated category $\C$. Especially, for a $2$-Calabi-Yau extriangulated category $\C$ with a cluster structure, we describe the cluster substructure in the cotorsion pairs. For rooted cluster algebras arising from $\C$ with cluster tilting objects, we give a one-to-one correspondence between cotorsion pairs in $\C$ and certain pairs of their rooted cluster subalgebras which we call complete pairs. Finally, we explain this correspondence by an example relating to a Grassmannian cluster algebra.

The first author constructed a q-parameterized spherical category $\sC$ over C(q) in [Liu15], whose simple objects are labelled by all Young diagrams. In this paper, we compute closed-form expressions for the fusion rule of $\sC$, using Littlewood-Richardson coefficients, as well as the characters (including a generating function), using symmetric functions with infinite variables.

Consider a complete abelian category which has an injective cogenerator. If its derived category is left-complete we show that the dual of this derived category satisfies Brown representability. In particular, this is true for the derived category of an abelian AB $4^{*}$ - $n$ category and for the derived category of quasi-coherent sheaves over a nice enough scheme, including the projective finitely dimensional space.

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