It takes two spectral sequences (Q6565652)

Consider the general linear Lie superalgebra \(\mathfrak{gl}(1|1)\) as matrices of the form \N\[\N\begin{pmatrix} c+ h/2 & x \\\Ny & c-h/2 \\\N\end{pmatrix}. \N\]\NGiven a \(\mathfrak{gl}(1|1)\)-module \(V\) on which \(h\) and \(c\) act semisimply, take invariants under \(c\) to obtain a vector superspace \(V^c\) with a supercommuting action of \(x\) and \(y\), along with a semisimple action of \(h\). The supercommuting actions of \(x\) and \(y\) allow one to write down a natural double complex whose terms are all \(M := V^c\). Thus consider the two spectral sequences associated to it. That is, the authors study the representation theory of the Lie superalgebra \(\mathfrak{gl}(1|1)\), constructing two spectral sequences which eventually annihilate precisely the superdimension 0 indecomposable modules in the finite-dimensional category. The pages of these spectral sequences, along with their limits, define symmetric monoidal functors on Rep \(\mathfrak{gl}(1|1)\). These two spectral sequences are related by contragredient duality, and from their limits, the authors construct explicit semisimplification functors, which they explicitly prove are isomorphic up to a twist. The authors use these tools to prove branching results for the restriction of simple modules over Kac-Moody and queer Lie superalgebras to \(\mathfrak{gl}(1|1)\)-subalgebras.