ScalingStacks

While Lie algebra representations and their tensor products have long played an important role in mathematics, their connection with low-dimensional topology is more recent. This involves quantum groups, which provide a deformation of the classical Lie theory. Reshetikhin–Turaev’s theory give rise to invariants of links and $3$-manifolds [ReTu].

Crane and Frenkel [CrFr] conjectured that there should be a “higher” representation theory where vector spaces are replaced by categories, and this would provide invariants of $4$-manifolds. The notion of higher representations was introduced first for type $A$ [ChRou] and then for general Kac–Moody algebras [Rou1, KhoLau]. In a work in preparation [Rou3], the second author gives a construction of a tensor product for higher representations of Kac–Moody algebras, in an $\infty$-categorical setting. An important feature is that the category underlying a tensor product of higher representations $\mathcal{V}$ and $\mathcal{V}^{\prime}$ of $\mathfrak{g}$ depends on the action of the positive part $\mathfrak{g}^{+}$ of $\mathfrak{g}$ on $\mathcal{V}$ and $\mathcal{V}^{\prime}$, and not just on the categories $\mathcal{V}$ and $\mathcal{V}^{\prime}$ themselves. Evidence for Crane and Frenkel’s program has also been provided by the work of Khovanov [Kho1], Webster [We] and others.

In this article, we consider the case of the super Lie algebra $\mathfrak{gl}(1|1)$. We do not discuss the notion of higher representations of $\mathfrak{gl}(1|1)$ (cf [Rou3]), but we focus on the positive part $\mathfrak{gl}(1|1)^{+}=\mathbb{C}e$, a one-dimensional odd super Lie algebra. The notion of a higher representation of $\mathfrak{gl}(1|1)^{+}$ is due to Khovanov [Kho]: it is the data of a differential category ${\mathcal{V}}$ over $\mathbb{F}_{2}$ together with a differential endofunctor $E$ and an endomorphism $\tau$ of $E^{2}$ with $d(\tau)=1$ satisfying $\tau^{2}=0$ and braid relations. So, a higher representation provides an endofunctor whose square is homotopic to $0$. An equivalent definition is that of an action of the monoidal category ${\mathcal{U}}$ generated by an object $E$ and a map $\tau:E^{2}\to E^{2}$ satisfying the conditions above.

We will allow a more general type of action where $E$ is given by a $({\mathcal{V}},{\mathcal{V}})$-bimodule.