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Three Results on Representations of Mackey Lie Algebras

  • Alexandru ChirvasituEmail author
Chapter
Part of the Developments in Mathematics book series (DEVM, volume 38)

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 VW-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 VW 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.

Key words

Mackey Lie algebra Finite length module Large annihilator Weight module Socle filtration 

Mathematics Subject Classification (2010):

17B10 17B20 17B65. 

Notes

Acknowledgements

I would like to thank Ivan Penkov and Vera Serganova for useful discussions on the contents of [1, 4] and for help editing the manuscript.

References

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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of WashingtonSeattleUSA

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