Three Results on Representations of Mackey Lie Algebras

Abstract

I. Penkov and V. Serganova have recently introduced, for any nondegenerate pairing $W \otimes V \rightarrow \mathbb{C}$ of vector spaces, the Lie algebra $\mathfrak{g}\mathfrak{l}^{M} = \mathfrak{g}\mathfrak{l}^{M}(V,W)$ consisting of endomorphisms of V whose duals preserve $W \subseteq V ^{{\ast}}$. In their work, the category $\mathbb{T}_{\mathfrak{g}\mathfrak{l}^{M}}$ of $\mathfrak{g}\mathfrak{l}^{M}$-modules, which are finite length subquotients of the tensor algebra $T(W \otimes V )$, is singled out and studied. Denoting by $\mathbb{T}_{V \otimes W}$ the category with the same objects as $\mathbb{T}_{\mathfrak{g}\mathfrak{l}^{M}}$ but regarded as V ⊗ W-modules, we first show that when W and V are paired by dual bases, the functor $\mathbb{T}_{\mathfrak{g}\mathfrak{l}^{M}} \rightarrow \mathbb{T}_{V \otimes W}$ taking a module to its largest weight submodule with respect to a sufficiently nice Cartan subalgebra of V ⊗ W is a tensor equivalence. Secondly, we prove that when W and V are countable-dimensional, the objects of $\mathbb{T}_{\mathrm{End}(V )}$ have finite-length as $\mathfrak{g}\mathfrak{l}^{M}$-modules. Finally, under the same hypotheses, we compute the socle filtration of a simple object in $\mathbb{T}_{\mathrm{End}(V )}$ as a $\mathfrak{g}\mathfrak{l}^{M}$-module.

This work was partially supported by the Danish National Research Foundation through the QGM Center at Aarhus University, and by the Chern-Simons Chair in Mathematical Physics at UC Berkeley.