Three Results on Representations of Mackey Lie Algebras

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


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. 



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.


  1. 1.
    E. Dan-Cohen, I. Penkov, and V. Serganova, A Koszul category of representations of finitary Lie algebras. ArXiv e-prints, May 2011.Google Scholar
  2. 2.
    M. Hazewinkel, Symmetric functions, noncommutative symmetric functions, and quasisymmetric functions. ArXiv Mathematics e-prints, October 2004.Google Scholar
  3. 3.
    G. W. Mackey, On infinite-dimensional linear spaces. Trans. Amer. Math. Soc., 57 (1945), 155–207.MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    I. Penkov and V. Serganova, Representation theory of Mackey Lie algebras and their dense subalgebras. This volume.Google Scholar
  5. 5.
    I. Penkov and K. Styrkas, Tensor representations of classical locally finite Lie algebras, in: Developments and trends in infinite-dimensional Lie theory, Progress in Mathematics, 288, Birkhäuser Boston, 2011, pp. 127–150.Google Scholar
  6. 6.
    S. V. Sam and A. Snowden, Stability patterns in representation theory. ArXiv e-prints, February 2013.Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of WashingtonSeattleUSA

Personalised recommendations