Generalized negligible morphisms and their tensor ideals

We introduce a generalization of the notion of a negligible morphism and study the associated tensor ideals and thick ideals. These ideals are defined by considering deformations of a given monoidal category $${\mathcal {C}}$$ C over a local ring R. If the maximal ideal of R is generated by a single element, we show that any thick ideal of $${\mathcal {C}}$$ C admits an explicitly given modified trace function. As examples we consider various Deligne categories and the categories of tilting modules for a quantum group at a root of unity and for a semisimple, simply connected algebraic group in prime characteristic. We prove an elementary geometric description of the thick ideals in quantum type A and propose a similar one in the modular case.

Deligne Categories and the Limit of Categories Rep(GL(m|n))

For each integer $t$ a tensor category $\mathcal{V}_t$ is constructed, such that exact tensor functors $\mathcal{V}_t\rightarrow \mathcal{C}$ classify dualizable $t$-dimensional objects in $\mathcal{C}$ not annihilated by any Schur functor. This means that $\mathcal{V}_t$ is the “abelian envelope” of the Deligne category $\mathcal{D}_t=\operatorname{Rep}(GL_t)$. Any tensor functor $\operatorname{Rep}(GL_t)\longrightarrow \mathcal{C}$ is proved to factor either through $\mathcal{V}_t$ or through one of the classical categories $\operatorname{Rep}(GL(m|n))$ with $m-n=t$. The universal property of $\mathcal{V}_t$ implies that it is equivalent to the categories $\operatorname{Rep}_{\mathcal{D}_{t_1}\otimes \mathcal{D}_{t_2}}(GL(X),\epsilon )$, ($t=t_1+t_2$, $t_1$ not an integer) suggested by Deligne as candidates for the role of abelian envelope.